EMOIRS

of the

American Mathematical Society

Number 1019

On First and Second Order

Planar Elliptic Equations

with Degeneracies

Abdelhamid Meziani

May 2012

•

Volume 217

•

Number 1019 (first of 4 numbers)

•

ISSN 0065-9266

American Mathematical Society

Number 1019

On First and Second Order

Planar Elliptic Equations

with Degeneracies

Abdelhamid Meziani

May 2012

• Volume 217 • Number 1019 (first of 4 numbers)

• ISSN 0065-9266

Library of Congress Cataloging-in-Publication Data

Meziani, Abdelhamid, 1957-

On ﬁrst and second order planar elliptic equations with degeneracies / Abdelhamid Meziani.

p. cm. — (Memoirs of the American Mathematical Society, ISSN 0065-9266 ; no. 1019)

“Volume 217, number 1019 (ﬁrst of 4 numbers).”
Includes bibliographical references.
ISBN 978-0-8218-5312-2 (alk. paper)
1. Degenerate diﬀerential equations.

2. Diﬀerential equations, Elliptic.

I. Title.

QA377.5.M49

2011

515

.3533—dc23

2011051781

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10 9 8 7 6 5 4 3 2 1

17 16 15 14 13 12

Contents

Introduction

Chapter 1.

Preliminaries

Chapter 2.

Basic Solutions

2.1.

Properties of basic solutions

2.2.

The spectral equation and Spec(

)

2.3.

Existence of basic solutions

2.4.

Properties of the fundamental matrix of (E

σ,

)

2.5.

The system of equations for the adjoint operator

∗

2.6.

Continuation of a simple spectral value

2.7.

Continuation of a double spectral value

2.8.

Purely imaginary spectral value

2.9.

Main result about basic solutions

Chapter 3.

Example

Chapter 4.

Asymptotic behavior of the basic solutions of

4.1.

Estimate of σ

4.2.

First estimate of φ and ψ

4.3.

End of the proof of Theorem 4.1

Chapter 5.

The kernels

5.1.

Two lemmas

5.2.

Proof of Theorem 5.1

5.3.

Modiﬁed kernels

Chapter 6.

The homogeneous equation

Lu = 0

6.1.

Representation of solutions in a cylinder

6.2.

Cauchy integral formula

6.3.

Consequences

Chapter 7.

The nonhomogeneous equation

Lu = F

7.1.

Generalized Cauchy Integral Formula

7.2.

The integral operator T

7.3.

Compactness of the operator T

Chapter 8.

The semilinear equation

Chapter 9.

The second order equation: Reduction

Chapter 10.

The homogeneous equation P u = 0

iii

CONTENTS

10.1.

Some properties

10.2.

Main result about the homogeneous equation P u = 0

10.3.

A maximum principle

Chapter 11.

The nonhomogeneous equation P u = F

Chapter 12.

Normalization of a Class of Second Order Equations with a

Singularity

Bibliography

Abstract

This paper deals with elliptic equations in the plane with degeneracies. The

equations are generated by a complex vector ﬁeld that is elliptic everywhere except
along a simple closed curve. Kernels for these equations are constructed. Properties
of solutions, in a neighborhood of the degeneracy curve, are obtained through inte-
gral and series representations. An application to a second order elliptic equation
with a punctual singularity is given.

Received by the editor June 17, 2010; and, in revised form, August 30, 2010.
Article electronically published on May 18, 2011; S 0065-9266(2011)00634-9.
2000 Mathematics Subject Classiﬁcation. Primary 35J70; Secondaries 35F05, 30G20.
Key words and phrases. CR equations, degenerate elliptic, spectral values, fundamental ma-

trix, asymptotic behavior, kernels, semilinear, normalization, vector ﬁelds.

Aﬃliation at time of publication: Department of Mathematics, Florida International Univer-

sity, Miami, Florida 33199; email: meziani@ﬁu.edu.

2011 American Mathematical Society

Introduction

This paper deals with the properties of solutions of ﬁrst and second order

equations in the plane. These equations are generated by a complex vector ﬁeld
X that is elliptic everywhere except along a simple closed curve Σ

⊂ R

. The

vector ﬁeld X is tangent to Σ and X

∧ X vanishes to ﬁrst order along Σ (and

so X does not satisfy H¨

ormander’s bracket condition). Such vector ﬁelds have

canonical representatives (see [8]). More precisely, there is a change of coordinates
in a tubular neighborhood of Σ such that X is conjugate to a unique vector ﬁeld L
of the form

(0.1)

L = λ

∂

∂t

− ir

∂

∂r

deﬁned in a neighborhood of the circle r = 0 in

R × S

, where λ

∈ R

+ i

R is an

invariant of the structure generated by X. We should point out that normalizations
for vector ﬁelds X such that X

∧ X vanishes to a constant order n > 1 along Σ

are obtained in [9], but we will consider here only the case n = 1. This canon-
ical representation makes it possible to study the equations generated by X in a
neighborhood of the characteristic curve Σ. We would like to mention a very recent
paper by F. Treves [13] that uses this normalization to study hypoellipticity and
local solvability of complex vector ﬁelds in the plane near a linear singularity. The
motivation for our work stems from the theory of hypoanalytic structures (see [12]
and the references therein) and from the theory of generalized analytic functions
(see [18]).

The equations considered here are of the form

Lu = F (r, t, u)

and

P u = G(r, t, u, Lu),

where P is the (real) second order operator

(0.2)

P = LL + β(t)L + β(t)L .

It should be noted that very little is known, even locally, about the structure of the
solutions of second order equations generated by complex vector ﬁelds with degen-
eracies. The paper [5] explores the local solvability of a particular case generated
by a vector ﬁeld of ﬁnite type.

An application to a class of second order elliptic operators with a punctual

singularity in

is given. This class consists of operators of the form

(0.3)

D = a

∂

∂x

+ 2a

∂

∂xy

+ a

∂

∂y

+ a

∂

∂x

+ a

∂

∂y

INTRODUCTION

where the coeﬃcients are real-valued, smooth, vanish at 0, and satisfy

≤

− a

+ y

)

≤ C

for some positive constants C

≤ C

. It turns out that each such operator D is

conjugate in U

\0 (where U is an open neighborhood of 0 ∈ R

) to a multiple of an

operator P given by (0.2).

Our approach is based on a thorough study of the operator

L given by

(0.4)

Lu = Lu − A(t)u − B(t)u .

For the equation

Lu = 0, we introduce particular solutions, called here basic solu-

tions. They have the form

w(r, t) = r

φ(t) + r

ψ(t) ,

where σ

∈ C and φ(t), ψ(t) are 2π-periodic and C-valued. Chapters 2 and 4

establish the main properties of the basic solutions. In particular, we show that
for every j

∈ Z, there are (up to real multiples) exactly two R-independent basic

solutions

(r, t) = r

±
j

(t) + r

±
j

(t)

with winding number j. For a given j, if σ

∈ C\R, then σ

−

= σ

; and if σ

∈ R

then we have only σ

−

≤ σ

. The basic solutions play a fundamental role in the

structure of the space of solutions of the equation

Lu = F .

In Chapter 6, we show that any solution of

Lu = 0 in a cylinder (0, R) × S

has a Laurent type series expansion in the w

’s. From the basic solutions of

L and

those of the adjoint operator

∗

, we construct, in Chapter 5, kernels Ω

and Ω

that allow us to obtain a Cauchy Integral Formula (Chapter 6)

(0.5)

u(r, t) =

∂

dζ

+ Ω

dζ

that represents the solution u of

Lu = 0 in terms of its values on the distinguished

boundary ∂

U = ∂U

\Σ.

For the nonhomogeneous equation

Lu = F , we construct, in Chapter 7, an

integral operator T , given by

(0.6)

T F =

−1

2π

F + Ω

dρdθ

This operator produces H¨

older continuous solutions (up to the characteristic circle

Σ), when F is in an appropriate L

-space. The properties of T allow us to estab-

lish, in Chapter 8, a similarity principle between the solutions of the homogeneous
equations

Lu = 0 and those of a semilinear equation Lu = F (r, t, u)

The properties of the (real-valued) solutions of P u = G are studied in Chapters

9 to 11. To each function u we associate a complex valued function w = BLu, called
here the L-gradient of u, and such that w solves an equation of the form

Lw = F .

The properties of the solutions of P u = G can thus be understood in terms of
the properties of their L-gradients. In particular series representations and integral
representations are obtained for u. A maximum principle for the solutions of P u = 0
holds on the distinguished boundary ∂

U , if the spectral values σ

satisfy a certain

INTRODUCTION

condition. In the last chapter, we establish the conjugacy between the operator D
and the operator P .

CHAPTER 1

Preliminaries

We start by reducing the main equation Lu = Au + Bu into a simpler form.

Then, we deﬁne a family of operators

, their adjoint

∗

, and prove a Green’s

formula. The operators

will be extensively used in the next chapter.

Let λ = a + ib

∈ R

+ i

∗

and deﬁne the vector ﬁeld L by

(1.1)

L = λ

∂

∂t

− ir

∂

∂r

For A

∈ C

(

C), with k ∈ Z

, set

2π

A(t)dt,

ν = 1

− Im

where for x

∈ R, [x] denotes the greatest integer less or equal than x. Hence,

∈ [0, 1). Deﬁne the function

m(t) = exp

it + i

t +

(A(s)

− A

)ds

Note that m(t) is 2π-periodic. The following lemma is easily veriﬁed.

Lemma

1.1. Let A, B

∈ C

(

C) and m(t) be as above. If u(r, t) is a solution

of the equation

(1.2)

Lu = A(t)u + B(t)u

then the function w(r, t) =

u(r, t)

m(t)

solves the equation

(1.3)

Lw = λ

− iν

w + C(t)w

where C(t) = B(t)

m(t)

In view of this lemma, from now on, we will assume that Re

= 0 and deal

with the simpliﬁed equation

(1.4)

Lu =

−iλνu + c(t)u

where ν

∈ [0, 1) and c(t) ∈ C

(

C).

Consider the family of vector ﬁelds

(1.5)

= λ

∂

∂t

− ir

∂

∂r

ABDELHAMID MEZIANI

where λ

= a + ib,

∈ R, and the associated operators L

deﬁned by

(1.6)

u(r, t) = λ

∂u

∂t

(r, t)

− ir

∂u

∂r

(r, t) + iλ

νu(r, t)

− c(t)u(r, t)

For

C-valued functions deﬁned on an open set U ∈ R

× S

, we deﬁne the bilinear

form

< f, g >= Re

f (r, t)g(r, t)

drdt

For the duality induced by this form, the adjoint of

(1.7)

∗

v(r, t) =

−

∂v

∂t

(r, t)

− ir

∂v

∂r

(r, t)

− iλ

νv(r, t) + c(t) v(r, t)

The function z

(r, t) =

|r|

is a ﬁrst integral of L

∗

× S

. That is, L

= 0,

= 0. Furthermore z

: R

× S

−→ C

∗

is a diﬀeomorphism. The following

Green’s identity will be used throughout.

Proposition

1.2. Let U

⊂ R

× S

be an open set with piecewise smooth

boundary. Let u, v

∈ C

(U ) with L

u and L

v integrable. Then,

(1.8)

∂U

=< u,

∗

v >

− < L

u, v > .

Proof.

Note that for a diﬀerentiable function f (r, t), we have

df =

−L

+ L

and

∧

2ia

∧ dt .

Hence,

∂U

(uL

v + vL

∧

−

− uL

∗

v + cvu

− ucv)

drdt

By taking the real parts, we get (1.8).

Remark

1.3. When b = 0 so that λ = a

∈ R

. The pushforward via the ﬁrst

integral r

reduces the equation

Lu = F into a Cauchy Riemann equation with

a singular point of the form

(1.9)

∂W

∂z

W +

B(t)

W + G(z).

Properties of the solutions of such equations are thoroughly studied in [10]. Many
aspects of CR equations with punctual singularities have been studied by a number
of authors and we would like to mention in particular the following papers [1], [14],
[15], [16] and [17].

Remark

1.4. We should point out that the vector ﬁelds involved here satisfy

the Nirenberg-Treves Condition (P) at each point of the characteristic circle. For
vector ﬁelds X satisfying condition (P), there is a rich history for the local solvability
of the

C-linear equation Xu = F (see the books [3], [12] and the references therein).

In [7], the semiglobal solvability of the equation P u = f is addressed, where P is
a pseudo-diﬀerential operator satisfying the Nirenberg-Treves Condition (P). Our

1. PRELIMINARIES

focus here is ﬁrst, on the semiglobal solvability in a tubular neighborhood of the
characteristic circle, and second, on the equations containing the term u which
makes them not

C-linear.

Remark

1.5. The operator

is invariant under the diﬀeomorphism Φ(r, t) =

(

−r, t) from R

× S

−

× S

. Hence, all the results about

stated in domains

contained in

× S

have their counterparts for domains in

−

× S

. Throughout

this paper, we will be mainly stating results for r

≥ 0.

CHAPTER 2

Basic Solutions

In this section we introduce the notion of basic solutions for

. We say that

w is a basic solution of

if it is a nontrivial solution of

w = 0, in

× S

, of

the form

(2.1)

w(r, t) = r

φ(t) + r

ψ(t),

with σ

∈ C and where φ(t), ψ(t) are 2π-periodic functions. These solutions play a

crucial role for the equations generated by L

. In a sense, they play roles similar

to those played by the functions z

in classical complex and harmonic analysis.

Consider, as our starting point, the basic solutions of

. These basic solutions

are known, since they can be recovered from those of equation (1.9) (see Remark
1.1). From

, we obtain the properties of the basic solutions for

. This is

done through continuity arguments in the study of an associated system of 2

× 2

ordinary diﬀerential equations in

with periodic coeﬃcients. By using analytic

dependence of the system with respect to the parameters, the spectral values σ of
the monodromy matrix can be tracked down. The main result (Theorem 2.1) states
that for every j

∈ Z, the operator L

has exactly two

R-independent basic solutions

with winding number j.

2.1. Properties of basic solutions

We prove that a basic solution has no vanishing points when r > 0 and that

one of its components φ or ψ is always dominating.

It is immediate, from (1.6), that in order for a function w(r, t), given by (2.1),

to satisfy

w = 0, the components φ and ψ need to be periodic solutions of the

system of ordinary diﬀerential equations

(2.2)

(t) = i(σ

− λ

ν)φ(t) + c(t)ψ(t)

(t) =

−i(σ − λ

ν)ψ(t) + c(t)φ(t) .

Note that if σ

∈ R, then w = r

(φ(t) + ψ(t)) and f = φ + ψ solves the equation

(2.3)

(t) = i(σ

− λ

ν)f (t) + c(t)f (t) .

Now we prove that a basic solution cannot have zeros when r > 0.

Proposition

2.1. Let w(r, t), given by (2.1), be a basic solution of

. Then

w(r, t)

= 0

∀(r, t) ∈ R

× S

Proof.

If σ

∈ R, we have w(r, t) = r

f (t) with f (t) satisfying (2.3).

w(r

, t

) = 0 for some r

> 0, then f (t

) = 0 and so f

≡ 0 by uniqueness of solutions

of the diﬀerential equation (2.3). Now, assume that σ = α + iβ with β

∈ R

∗

ABDELHAMID MEZIANI

Suppose that w is a basic solution and w(r

, t

) = 0 for some (r

, t

)

∈ R

× S

Consider the sequence of real numbers r

= r

exp(

−kπ/|β|) with k ∈ Z

. Then

−→ 0 as k −→ ∞ and r

2iβ

= r

2iβ

. It follows at once from w(r

, t

) = 0 and (2.1)

that w(r

, t

) = 0 for every k

∈ Z

. Note that from (2.1) we have

|w(r, t)| ≤ Er

where E = max(

|φ(t)| + |ψ(t)|). Note also that since L

is elliptic in

× S

, then

the zeros of any solution of the equation

u = 0 are isolated in

× S

The pushforward via the mapping z = r

of the equation

w = 0 in

×S

is the singular CR equation

∂W

∂z

νe

2iθ

2az

−

C(z)e

2iθ

2iaz

where W (z) and C(z) are the pushforwards of w(r, t) and c(t) and where θ is the
argument of z. We are going to show that W has the form W (z) = H(z) exp(S(z))
where H is holomorphic in the punctured disc D

∗

(0, R), S(z) continuous in D

∗

(0, R)

and satisﬁes the growth condition

|S(z)| ≤ log

|z|

for some positive constants K

and p. For this, consider the function M (z) deﬁned by

M (z) =

νe

2iθ

−

C(z)e

2iθ

2ia

W (z)

for 0 <

|z| < R, W (z) = 0 and by M(z) = 1 on the set of isolated points where

W (z) = 0. This function is bounded and it follows from the classical theory of CR
equations (see [2] or [18]) that

N (z) =

−1

D(0,R)

M (ζ)

− z

dξdη

(ζ = ξ + iη) is continuous, satisﬁes

∂N (z)

∂z

= M (z) and

|N(z

)

− N(z

)

| ≤ A||M||

∞

− z

| log

− z

∀z

, z

∈ D(0, R)

for some positive constant A. Deﬁne S by S(z) =

N (z)

− N(0)

. We have then, for

= 0,

∂S

∂z

(z)

W (z)

and

|S(z)| ≤ B log

|z|

with B = A

||M||

∞

. Let H(z) = W (z) exp(

−S(z)). Then H is holomorphic in

0 <

|z| < R and it satisﬁes

|H(z)| ≤ |W (z)| exp(|S(z)|) ≤ |W (z)|

(2R)

|Z|

≤ C

|z|

for some constants C

and s

∈ R. The last inequality follows from the estimate

|w| ≤ Er

. This means that the function H has at most a pole at z = 0. Since

w(r

, t

) = 0, then H(z

) = 0 for every k and z

= r

−→ 0. Hence H ≡ 0

and w

≡ 0 which is a contradiction.

Corollary

2.2. If w = r

φ(t)+r

ψ(t) is a basic solution of

with σ = α+iβ

and β

= 0, then for every t ∈ R, |φ(t)| = |ψ(t)|.

2.2. THE SPECTRAL EQUATION AND Spec(

)

Proof.

By contradiction, suppose that there is t

∈ R such that |φ(t

)

| =

|ψ(t

)

|. Let x

∈ R such that ψ(t

) =

−e

φ(t

). Then the positive number

= exp(x

/2β) satisﬁes r

iβ

= r

−iβ

and consequently,

w(r

, t

) = r

iβ

φ(t

) + r

iβ

ψ(t

)) = 0.

This contradicts Proposition 2.1.

This corollary implies that, for a given basic solution w = r

φ + r

ψ with

∈ C\R, one of the functions φ or ψ is dominant. That is, |φ(t)| > |ψ(t)| or

|ψ(t)| > |φ(t)| for every t ∈ R. Hence the winding number of w, Ind(w) is well
deﬁned and we have Ind(w) = Ind(φ) if

|φ| > |ψ| and Ind(w) = Ind(ψ) otherwise.

When σ

∈ R, we have w = r

f (t) with f nowhere 0 and so Ind(w) = Ind(f ).

For a basic solution w = r

φ + r

ψ with

|φ| > |ψ|, we will refer to σ as the

exponent of w (or a spectral value of

) and deﬁne the character of w by

Char(w) = (σ, Ind(w)).

We will denote by Spec(

) the set of exponents of basic solutions. That is,

(2.4)

Spec(

) =

{σ ∈ C; ∃w, Char(w) = (σ, Ind(w))}, .

Remark

2.3. When σ

∈ C\R and w = r

φ(t) + r

ψ(t) is a basic solution with

Char(w) = (σ, Ind(φ)), the function

w = r

(iφ(t)) + r

iψ(t) is also a basic solution

with Char(w) = Char(

w) and w,

w are

R-independent.

When σ = τ

∈ R, and w = r

f (t) is a basic solution with Char(w) =

(τ, Ind(f )), it is not always the case that there is a second

R-independent basic

solution with the same exponent τ . There is however a second

R-independent basic

solution

w = r

g(t) with the same winding number (Ind(f ) = Ind(g)) but with a

diﬀerent exponent τ

(see Proposition 2.6).

The following proposition follows from the constancy of the winding number

under continuous deformations.

Proposition

2.4. Let w

(r, t) = r

σ()

φ(t, )+r

σ()

ψ(t, ) be a continuous family

of basic solutions of

with

∈ I, where I ⊂ R is an interval. Then Char(w

)

depends continuously on and Ind(w

) is constant.

2.2. The spectral equation and Spec(

)

We use the 2

×2 system of ordinary diﬀerential equations to obtain an equation

for the spectral values in terms of the monodromy matrix. Results about the CR
equation (1.9) are then used to list the properties of Spec(

In order for a function

w(r, t) = r

φ(t) + r

ψ(t)

to be a basic solution of

, the 2π-periodic and

-valued function V (t) =

φ(t)
ψ(t)

must solve the periodic system of diﬀerential equations

σ,

)

V = M(t, σ, )V

ABDELHAMID MEZIANI

where

M(t, σ, ) =

⎛
⎜

⎜

⎝

− λ

c(t)

−i

− λ

⎞
⎟

⎟

⎠ .

Note that since M is linear in σ

∈ C and real analytic in ∈ R, then any solution

V (t, σ, ) is an entire function in σ and real analytic in . The fundamental matrix
of (E

σ,

) is the 2

× 2 matrix V(t, σ, ) satisfying

V = M(t, σ, )V,

V(0, σ, ) = I

where I is the identity matrix. We know from Floquet theory that

V(t, σ, ) = P(t, σ, ) exp(tK(σ, ))

where P is a 2π-periodic matrix (in t) and P and K are entire in σ and real analytic
in . The monodromy matrix of (E

σ,

) is

B(σ, ) = V(2π, σ, ) = exp(2πK(σ, )) .

The Liouville-Jacobi formula gives

det(V(t, σ, )) = exp

tr(M(s, σ, ))ds

= exp

|λ

σt

where det(A) and tr(A) denote the determinant and the trace of the matrix A.
Hence,

(2.5)

det(B(σ, )) = exp

4πb
|λ

In order for system (E

σ,

) to have a periodic solution, the corresponding monodromy

matrix B must have 1 as an eigenvalue. Thus σ must solve the spectral equation

− tr(B(σ, )) + det(B(σ, )) = 0.

or equivalently, F (σ, ) = 0, where

(2.6)

F (σ, ) = tr(B(σ, ))

− 1 − exp

4πb
|λ

We ﬁrst verify that Spec(

) is a discrete set.

Lemma

2.5. For every

∈ R, Spec(L

) is a discrete subset of

Proof.

By contradiction, suppose that there exists

∈ R such that Spec(L

)

has an accumulation point in

C. This means that the roots of the solutions of the

spectral equation F (σ,

) = 0 have an accumulation point. Since F is an entire

function, then F (σ,

)

≡ 0. Thus, Spec(L

) =

C. Let

φ(t, σ)
ψ(t, σ)

be a continuous

family of periodic solutions of (E

σ,

). By Proposition 2.4, we can assume that

|φ| > |ψ| for every σ ∈ R

+ i

R and that Ind(φ) = j

(is constant). Now the ﬁrst

equation of (E

σ,

) gives

2πi

2π

φ(t, σ)

dt = σ

− λ

ν +

2πi

2π

c(t)

ψ(t, σ)

φ(t, σ)

dt .

2.3. EXISTENCE OF BASIC SOLUTIONS

That is,

σ = λ

+ ν)

−

2πi

2π

c(t)

ψ(t, σ)

φ(t, σ)

∀σ ∈ R

+ i

R .

This is a contradiction since

< |c|.

The following proposition describes the spectrum of

Proposition

2.6. For every j

∈ Z, there exist τ

∈ R with τ

−

≤ τ

and

∈ C

k+1

(

C), such that w

(r, t) = r

(t) are

R-independent basic solution

with

Char(w

) = (τ

, j) .

Furthermore, Spec(

) =

{τ

, j

∈ Z},

· · · < τ

−

−1

≤ τ

−1

< τ

−

≤ τ

< τ

−

≤ τ

· · ·

with

lim

→−∞

−∞, lim

→∞

∞.

Proof.

We have here λ

= a > 0. The pushforward of the equation

w = 0

via the ﬁrst integral z = r

of L

gives a CR equation with a singularity of the

form studied in [10]. The spectral values τ

of the CR equations are as in the

proposition. It remains only to verify that

(or its equivalent CR equation) has

no complex spectral values. The Laurent series representation for solutions of the
CR equation (see [10]) imply that any solution of

w = 0 can be written as

w(r, t) =

∈Z

−

(t) + c

+
j

(t)

with c

∈ R. Now, if w = r

φ(t) + r

ψ(t) is a basic solution of

, then it follows

at once, from the series representation, that σ is one of the τ

’s.

2.3. Existence of basic solutions

We use the spectral equation together with Proposition 2.6 to show the exis-

tence of basic solutions for

with any given winding number. More precisely, we

have the following proposition.

Proposition

2.7. For every j

∈ Z, there exists σ

()

∈ Spec(L

) such that

() depends continuously on

∈ R, σ

(0) = τ

, and the corresponding basic

solution

(r, t, ) = r

±
j

()

φ(t, ) + r

±
j

()

ψ(t, )

is continuous in and Char(w

) = (σ

(), j).

Proof.

For a given j

∈ Z, it follows from Proposition 2.6 that the monodromy

matrix B(τ

, 0) admits 1 as an eigenvalue. Since the spectral function F (σ, ) given

by (2.6) is entire in

C × R and since F (τ

, 0) = 0, then F (σ, ) = 0 deﬁnes an

analytic variety

V in C × R passing through the points (τ

, 0). The variable can

be taken as a parameter for a branch of

V through the point (τ

, 0). This means

that the equation F (σ, ) = 0 has a solution σ = g()

∈ C, with g continuous and

ABDELHAMID MEZIANI

g(0) = τ

. In fact, g is real analytic except at isolated points. The matrix B(g(), )

is continuous and has 1 as an eigenvalue for every . Let E

be an eigenvector of

B(τ

, 0) with eigenvalue 1. We can select a continuous vector E

()

∈ C

such

that

B(g(), )E

() = E

()

and

(0) = E

Let V (t, ) = V(t, g(), )E

(). Then V (t, ) is a periodic solution of the equation

g(),

). If we set V (t, ) =

(t, )

, then

w(r, t, ) = r

g()

(t, ) + r

g()

(t, )

is a basic solution of

and it depends continuously on . Since for = 0, w(r, t, 0)

has character (τ

, j), then by Proposition 2.4, the character of w(r, t, ) is either

(g(), j) if

|φ

| > |ψ

| or (g(), j) if |φ

| < |ψ

|. In the ﬁrst case, σ

() = g()

∈

Spec(

) and, in the second, σ

() = g()

∈ Spec(L

2.4. Properties of the fundamental matrix of (E

σ,

)

We prove some symmetry properties of the fundamental matrix and of the

monodromy matrix that will be used shortly.

Proposition

2.8. There exist functions f (t, σ, s), g(t, σ, s) of class C

k+1

∈ R, analytic in (σ, s) ∈ C × R, such that the fundamental matrix V(t, σ, ) of

σ,

) has the form

(2.7)

V(t, σ, ) =

f (t, σ,

)

g(t, σ,

)

g(t, σ,

)

f (t, σ,

)

exp

|λ

Furthermore, f and g satisfy

(2.8)

f (t, σ,

)f (t, σ,

)

− |λ

g(t, σ,

)g(t, σ,

)

≡ 1.

Proof.

If we use the substitution V = Z exp

bσt

|λ

in equation (E

σ,

), then

the system for Z is

(2.9)

Z = A(t, σ,

with

A(t, σ,

) =

iμ

d(t,

)

d(t,

)

−iμ

and where

μ =

aσ

+ b

− ν , and d(t,

) =

c(t)

+ b

The fundamental matrix Z(t, σ,

) of (2.9) with Z(0, t,

) = I is therefore of class

k+1

in t and analytic in (σ, s) with s =

For functions F (t, μ,

) and G(t, μ,

), with μ

∈ R, we use the notation

and

2.4. PROPERTIES OF THE FUNDAMENTAL MATRIX OF (E

σ,

)

With this notation, system (2.9) has the form

(2.10)

Z = (D

iμ

+ J

) Z.

Note that we have the following relations

= D

= J

F G

= J

F G

= D

|λ

The fundamental matrix Z is obtained as the limit, Z = lim

→∞

, where the

matrices Z

(t, σ, ) are deﬁned inductively by Z

= I and

k+1

(t, σ, ) = I +

iμ

+ J

) Z

(s, σ, )ds.

Now, we prove by induction that Z

= D

+ J

, where F

and G

are polyno-

mials in the variable μ, analytic and even in the variable . The claim is obviously
true for Z

= I. Suppose that Z

has the desired property for k = 0,

· · · , n, then

n+1

(t, σ, )

= I +

iμ

+ J

) (D

+ J

) ds

= D

n+1

+ J

n+1

where

n+1

(t, μ,

)

= 1 +

iμF

(s, μ,

) + c(s)G

(s, μ,

)

n+1

(t, μ,

)

−

iμG

(s, μ,

) + c(s)F

(s, μ,

)

ds.

By taking the limit as k

→ ∞, we get the fundamental matrix

(2.11)

Z(t, μ, ) = D

(t,μ,

)

+ J

(t,μ,

)

where F

and G

are entire functions with respect to the real parameters μ and

. The fundamental matrix Z(t, σ, ) of (2.9) is therefore D

f (t,σ,

)

+ J

g(t,σ,

)

where f and g are the holomorphic extensions of F

and G

(obtained by replacing

μ by

aσ

+ b

− ν, with σ ∈ C). The proposition follows immediately, since

V = Z exp

bσt

|λ

A direct consequence of expression (2.7) of the proposition is the following:

Proposition

2.9. Let V(t, σ, ) be the fundamental matrix of (E

σ,

), then the

fundamental matrix of equation (E

σ,

) is

(2.12)

V(t, σ, ) = JV(t, σ, )J

where J =

The monodromy matrix of (E

σ,

) has the form

(2.13)

B(σ, ) =

p(σ,

)

q(σ,

)

q(σ,

)

p(σ,

)

exp

2πb

+ b

with p(σ,

) = f (2π, σ,

), q(σ,

) = g(2π, σ,

) satisfying

(2.14)

p(σ,

)p(σ,

)

− |λ

q(σ,

)q(σ,

)

≡ 1.

ABDELHAMID MEZIANI

Note that

(2.15)

det(B(σ, )) = exp

4πb

+ b

We denote by Spec(B(., )) the set of spectral values of B(., ), i.e.

Spec(B(., )) =

{σ ∈ C; det(B(σ, ) − I) = 0}.

It follows from (2.13) and Proposition 2.9 that

(2.16)

Spec(B(., )) = Spec(B(.,

−)) = Spec(B(., )).

An element σ

∈ Spec(B(., )) is said to be a simple (or a double) spectral value if

the corresponding eigenspace has dimension 1 (or 2).

The spectral function F deﬁned in (2.6) takes form

(2.17)

F (σ,

) = p(σ,

) + p(σ,

)

− 2 cosh

2πb

+ b

Thus if F (σ,

) = 0, then F (σ, ) = 0. We have therefore

Corollary

2.10. If σ

∈ Spec(L

), then σ or σ

∈ Spec(L

−

2.5. The system of equations for the adjoint operator

∗

The properties of the fundamental matrix of system E

σ,

will be used to obtain

those for the adjoint operator. The system of ordinary diﬀerential equations for the
adjoint operator

∗

given in (1.7) is

(

μ,

)

V =

M(t, μ, )V

where

(2.18)

M(t, μ, ) =

⎛
⎜

⎜

⎝

μ + λ

−

c(t)

−

c(t)

−i

μ + λ

⎞
⎟

⎟

⎠ .

Thus, if V (t) =

X(t)
Z(t)

is a periodic solution of (

μ,

), then

w(r, t) = r

X(t) + r

Z(t)

is a basic solution of

∗

The relation between the fundamental matrices of this system and those for

σ,

is given by the following proposition.

Proposition

2.11. The fundamental matrix of (

μ,

) is

(2.19)

V(t, μ, ) = DV(t,

−μ, −)D

where V(t,

−μ, −) is the fundamental matrix of (E

−μ,−

) and D =

−1

2.6. CONTINUATION OF A SIMPLE SPECTRAL VALUE

Proof.

If V =

X
Z

solves (

μ,

), then DV =

−Z

solves the equa-

tion (E

−μ,−

). Therefore, if

is a fundamental matrix of (

μ,

), then

−X

−Z

is a fundamental matrix of (E

−μ,−

Immediate consequences are the following corollaries.

Corollary

2.12. The monodromy matrix of (

μ,

) is

(2.20)

B(μ, ) = DB(

−μ, −)D

where B(σ, ) is the monodromy matrix of (E

σ,

). Furthermore, if σ

∈ Spec(B(., ))

and B(σ, )E = E, then

−σ ∈ Spec(

B(.,

−)) and

−σ, −)DE = DE.

Corollary

2.13. If σ

∈ Spec(L

), then either

−σ ∈ Spec(L

∗

) or

−σ ∈

Spec(

∗

2.6. Continuation of a simple spectral value

We start from a simple spectral value, when = 0, and use the properties of

the fundamental matrix to obtain the behavior of σ() for near 0.

Proposition

2.14. Suppose that τ

∈ Spec(B(., 0)) and that τ is simple. Then

there exist δ > 0 and a unique function σ

∈ C

([

−δ, δ], R) such that σ(0) = τ and

σ()

∈ Spec(B(., )) for every ∈ [−δ, δ].

Proof.

The matrix B(τ, 0) has a single eigenvector U (up to a multiple) with

eigenvalue 1. Since det(B(τ, 0)) = 1 (see (2.13) and (2.14)), then B(τ, 0) is similar

to the matrix

. Let V(t, σ, ) be the fundamental matrix of (E

σ,

). The

function

φ(t)
ψ(t)

= V(t, τ, 0)U,

generates all periodic solutions of (E

τ,0

). First, we show that we can ﬁnd a generator

of the form

f (t)
f (t)

for some function f . For this, note that since λ

= a

∈ R,

then it follows from (2.2) that

ψ(t)
φ(t)

is also a periodic solution of (E

τ,0

). Hence,

there exists c

∈ C, |c| = 1 such that ψ(t) = cφ(t) and φ(t) = cψ(t). If c = 1, then

we can take f = φ, if c

= −1, we can take f = φ + ψ, and if c = −1, we take

f = iφ. The vector U

f (0)
f (0)

is the eigenvector of B(τ, 0) that generates the

solution

f (t)
f (t)

We know from Proposition 2.7 that the spectral function F (σ,

) given in (2.6)

has a root σ() with σ(0) = τ . Furthermore, σ() is real analytic in a neighborhood
of 0

∈ R, except possibly at = 0. Now we show that there is only one such function

in a neighborhood of 0 and that it is real-valued. Starting from U

, we can ﬁnd a

ABDELHAMID MEZIANI

continuous vector U ()

∈ C

with U (0) = U

and such that B(σ(), )U () = U ().

The function

φ(t, )
ψ(t, )

= V(t, σ(), )U ()

is a periodic solution of (E

σ().

) such that φ(t, 0) = f (t) and ψ(t, 0) = f (t). It

follows from Corollary 2.13 that

−τ ∈ Spec(

B(., 0)) and

−σ() ∈ Spec(

B(., )).

Note that if V (t) is a periodic solution of (E

τ,0

), then DV (t) is a periodic solution

of (

−τ,0

). Thus,

f (t)

−f(t)

solves (

−τ,0

). Let U

()

∈ C

be a continuous

eigenvector of B(σ(),

−) such that U

(0) = U

. Set U

() = DU

(). Then, it

follows from Corollary 2.12, that

−σ(), )U

() = DB(σ(),

−)DDU

() = DU

() = U

().

Therefore,

X(t, )
Z(t, )

V(t,

−σ(), )U

()

is a periodic solution of (

−σ().

) with X(t, 0) = f (t) and Z(t, 0) =

−f(t).

The corresponding basic solutions of

and

∗

are respectively,

(r, t) = r

σ()

φ(t, ) + r

σ()

ψ(t, ),

and

(r, t) = r

−σ()

X(t, ) + r

−σ()

Z(t, ).

We apply Green’s formula (1.8) to the pair w

, w

in the cylinder A = [R

, R

]

×S

(with 0 < R

< R

) to get

∂A

(r, t)w

(r, t)

= 0 .

That is,

(2.21)

2π

−σ

− R

−σ

)φX + (R

−σ

− R

−σ

)ψZ

idt

= 0 .

Suppose that σ() is not

R-valued in a neighborhood of 0. Then σ() = α()+iβ()

with β() > 0 (or < 0) in a an interval (0,

). If we set p = log R

and q = log R

we get

−σ

− R

−σ

= e

2iβp

− e

2iβq

= 2i sin(β(p

− q))e

iβ(p+q)

and (2.21) becomes (with x = p + q arbitrary)

(2.22)

2π

iβx

φ(t, )X(t, )

− ie

−iβx

ψ(t, )Z(t, )

= 0.

Let

P () + iQ() =

2π

φ(t, )X(t, )dt and R() + iS() =

2π

ψ(t, )Z(t, )dt .

From (2.22), we have

cos(β()x)(P ()

− R()) − sin(β()x)(Q() + S()) = 0,

∀x ∈ R .

Therefore,

P ()

− R() = 0, Q() + S() = 0,

∀ ∈ (0,

2.7. CONTINUATION OF A DOUBLE SPECTRAL VALUE

By continuity, we get P (0) = R(0) and Q(0) =

−S(0). But,

P (0) + iQ(0)

2π

φ(t, 0)X(t, 0)dt =

2π

|f(t)|

R(0) + iS(0)

2π

ψ(t, 0)Z(t, 0)dt =

−

2π

|f(t)|

and it follows from P (0) = R(0) that

2π

|f|

dt = 0. This is a contradiction since

= 0. This means that σ() is an R-valued function in a neighborhood of = 0.

Now we show that σ() is unique near = 0. By contradiction, suppose that

there is another real valued solution σ

(), with σ() < σ

() in an interval (0,

and σ(0) = σ

(0) = τ .

Let φ(t, ) and ψ(t, ) be as above.

Let U

() be an

eigenvector (with eigenvalue 1) of B(σ

(),

−) such that U

(0) =

−iU

, where U

is the eigenvector used above. Let U

() = DU

(). Then

−σ

(), )U

() = U

()

and

(0) = iDU

To U

() corresponds the 2π-periodic solution

(t, )

V(t,

−σ

(), )U

()

of (

−σ

().

) with X

(t, 0) = i f (t) and Z

(t, 0) =

−if(t). The corresponding basic

solution of

∗

(r, t) = r

−σ

()

(t, ) + Z

(t, )).

The Green’s formula, applied to the pair w

, w

in the cylinder (R

, R

)

× S

gives

2π

−σ

− R

−σ

)(φ + ψ)(X

+ Z

)idt

= 0 .

Thus,

2π

(φ(t, ) + ψ(t, ))(X

(t, ) + Z

(t, ))idt

= 0

∀ ∈ (0,

) .

By letting

→ 0, we get again

2π

|f(t)|

dt = 0, which is a contradiction. This

shows that σ() is unique for near 0.

2.7. Continuation of a double spectral value

This time we study the behavior of σ() when σ(0) has multiplicity 2. Hence

assume that τ

∈ Spec(B(., 0)) has multiplicity 2. Therefore, B(τ, 0) = I. We start

with the following proposition.

Proposition

2.15. If B(τ, 0) = I, then

∂trB

∂σ

(τ, 0) = 0 and

∂

trB

∂σ

(τ, 0)

= 0.

The proof of this proposition makes use of the the following lemma.

Lemma

2.16. Given M > 0, there is a positive constant C such that

(1 + 2x)(1 + 2y)

− 4

xy(1 + x)(1 + y)

≥ C,

∀x, y ∈ [0, M].

ABDELHAMID MEZIANI

Proof.

Consider the function g(x, y) = (1 + 2x)

(1 + 2y)

−16xy(1+x)(1+y).

It can be easily veriﬁed that g(x, y)

≥ 1 in the square [0, M]

. This implies in turn

that

(1 + 2x)(1 + 2y)

− 4

xy(1 + x)(1 + y)

≥

1 + 8M + 8M

Proof of Proposition

2.15. Let V(t, σ, ) be the fundamental matrix of

equation (E

σ,

) given by (2.7). Its derivative V

, with respect to σ, satisﬁes the

system

(2.23)

= MV

+ M

(0, σ, ) = 0

(the last condition follows from V(0, σ, ) = I). Note that

= D

i/λ

⎛
⎜

⎝

−i

⎞
⎟

⎠ .

We consider (2.23) as a nonhomogeneous system in V

and we get

(2.24)

(t, σ, ) = V(t, σ, )

V(s, σ, )

−1

i/λ

V(s, σ, )ds .

By using formula (2.7), we have

−1

i/λ

V = i

where

f (t, σ,

)f (t, σ,

)

+ λ

g(t, σ,

)g(t, σ,

)

f (t, σ,

)g(t, σ,

)

−

f (t, σ,

)g(t, σ,

)

−

f (t, σ,

)f (t, σ,

)

− λ

g(t, σ,

)g(t, σ,

In particular

(2.25)

tr(V

−1

i/λ

V)(t, σ, 0) = iN

(t, σ, 0) + iN

(t, σ, 0)

≡ 0.

If we set t = 2π in (2.24), we get

(2.26)

∂B

∂σ

(σ, ) = B(σ, )

2π

V(s, σ, )

−1

i/λ

V(s, σ, )ds.

Since, B(τ, 0) = I, then it follows at once from (2.25) and (2.26) that

∂trB

∂σ

(τ, 0) =

0. Now we compute V

σσ

. We have

(2.27)

σσ

= MV

σσ

+ 2D

i/λ

σσ

(0, σ, ) = 0

2.7. CONTINUATION OF A DOUBLE SPECTRAL VALUE

and after integrating this nonhomogenous system and using (2.24), we obtain

(2.28)

σσ

(t, σ, )

= 2V(t, σ, )

−1

i/λ

(s, σ, )ds

= 2V(t, σ, )

L(s, σ, )L(u, σ, )duds

where

L(t, σ, ) = V

−1

i/λ

V(t, σ, ) .

We have in particular

(2.29)

L(t, σ, 0) =

P (t, σ)

2Q(t, σ)

−2Q(t, σ) −P (t, σ)

where

(2.30)

P (t, σ) = f (t, σ, 0)f (t, σ, 0) + a

g(t, σ, 0)g(t, σ, 0)

Q(t, σ) = af (t, σ, 0)g(t, σ, 0).

If we set t = 2π in (2.28), we get

σσ

(σ, ) = 2B(σ, )

2π

L(s, σ, )L(u, σ, )duds

and since B(τ, 0) = I, we have

(2.31)

∂

tr(B)

∂σ

(τ, 0) = 2

2π

tr(L(s, τ, 0)L(u, τ, 0))duds

It follows from (2.29) that

(2.32)

−a

tr(L(s, τ, 0)L(u, τ, 0)) = P (s, τ )P (u, τ )

− 4Re

Q(s, τ )Q(u, τ )

Let g(t, τ, 0) =

ρ(t)

iβ(t)

(thus, ρ = a

|g| and β is the argument of g) and then since

f and g satisfy (2.8) we have f (t, τ, 0) = (1 + ρ(t)

)

1/2

iα(t)

. With this notation,

the functions P and Q become

P (t, τ ) = 1 + 2ρ(t)

and

Q(t, τ ) = ρ(t)

1 + ρ(t)

i(α(t)+β(t))

If we set x = ρ(s)

, y = ρ(u)

, and θ = α(s) + β(s)

− α(u) − β(u), then formula

(2.32) becomes

(2.33)

−a

tr(L(s, τ, 0)L(u, τ, 0)) = (1 + 2x)(1 + 2y)

− 4

xy(1 + x)(1 + y) cos θ.

Since, x, y are positive and bounded (g is bounded over the interval [0, π]), then
Lemma 2.16 implies that there is a positive constant C such that

tr(L(s, τ, 0)L(u, τ, 0))

≤ −C

∀u, s ∈ [0, 2π].

Therefore, by (2.31), we have tr(B

σσ

(τ, 0)

= 0. This completes the proof of the

Proposition.

The behavior of the spectral values of

is given by the following proposition.

Proposition

2.17. Suppose that B(τ, 0) = I. Then there exists

> 0 such

that the spectral values of

through τ satisfy one of the followings:

(1) there is a unique continuous function σ() deﬁned in [

−

] such that

σ()

∈ Spec(L

), σ()

∈ C\R for = 0 and σ(0) = τ;

ABDELHAMID MEZIANI

(2) there are two continuous

R-valued functions σ

(), σ

() deﬁned in

[

−

], such that σ

(), σ

()

∈ Spec(L

), σ

() < σ

() for

= 0,

and σ

(0) = σ

(0) = τ

(3) there is a unique continuous

R-valued function σ() deﬁned in [−

]

such that σ()

∈ Spec(L

) and σ(0) = τ .

Proof.

It follows from Proposition 2.15 that the spectral function F (σ, )

deﬁned in (2.6) satisﬁes

F (τ, 0) = 0,

∂F

∂σ

(τ, 0) = 0,

and

∂

∂σ

(τ, 0)

= 0 .

Since F is analytic in both variables, then by the Weierstrass Preparation Theorem
(see [6]) we can ﬁnd analytic functions G(σ, ), A

() and A

() with (σ, ) near

(τ, 0)

∈ C × R, such that G(τ, 0) = 0, A

(0) = A

(0) = 0 and

F (σ, ) = G(σ, )

(σ

− τ)

− 2A

()(σ

− τ) + A

()

Thus, there exists

> 0 such that the roots of the spectral equation F (σ, ) = 0

in a neighborhood of (τ, 0) are given by the quadratic formula

1,2

() = τ + A

()

− A

() .

The conclusion of the proposition follows depending on the sign of the discriminant
A

− A

2.8. Purely imaginary spectral value

We study here the behavior of the monodromy matrix at possible spectral value

on the imaginary axis.

Proposition

2.18. Suppose that for some

∈ R

∗

, the operator

has a

spectral value σ

of the form

(2.34)

= i

|λ

with

∈ Z

∗

Then the monodromy matrix B(σ

) is similar to

Proof.

If σ

given by (2.34) is a spectral value, then det B(σ

) = 1, by

(2.5). Hence 1 is an eigenvalue of B(σ

) and, a priori, it could have multiplicity

2. In which case B(σ

) = I. We are going to show that this case does not happen.

By contradiction, suppose that B(σ

) = I. First we prove that tr(B

(σ

))

0. From formulas (2.26), (2.25), and (2.8) we have

∂tr(B)

∂σ

(σ

) =

2π

m(s)ds

where

m(s)

− λ

|λ

f (s, σ

)f (s, σ

)

− |λ

g(s, σ

)g(s, σ

)

−2ib

|λ

2.8. PURELY IMAGINARY SPECTRAL VALUE

This shows that tr(B

(σ

))

= 0. Hence, it follows that the spectral function

satisﬁes

∂F

∂σ

(σ

) =

∂tr(B)

∂σ

(σ

)

= 0 .

By the implicit function theorem, the germ of the analytic variety F (σ, ) = 0
through (σ

) is smooth and there is a unique analytic function σ() deﬁned near

with σ(

) = σ

such that F (σ(), )

≡ 0. It follows that B(., ) has a unique

spectral value through (σ

Let U

() be a continuous eigenvector (with eigenvalue 1) of B(σ(), ) deﬁned

in an interval (

− δ,

+ δ) for some δ > 0. We can assume that U

(

) =

α
β

with α

= 0. Now, consider the equation

G(σ, , z) = (B(σ, )

− I)

z
1

= 0

in a neighborhood of (σ

, 0)

∈ C × R × C. This equation deﬁnes a germ of an

analytic variety of real dimension 1 in

C × R × C that passes through the point

(σ

, 0). Therefore, there exists δ

> 0 and continuous functions σ

() and z()

deﬁned in [

− δ

+ δ

] such that σ

(

) = σ

, z(

) = 0 and G(σ

(), , z()

≡ 0.

The continuous vector U

() =

z()
1

is therefore an eigenvector with eigenvalue

1 of B(σ

(), ). Moreover, U

() and U

() are independent for close to

. By

the uniqueness of the spectral value established above, we get σ

() = σ(). This

means that B(σ(), )

≡ I for close to

. From this and (2.5) we get that

det(B(σ(), )) = exp

4πb
|λ

σ()

≡ 1

and therefore, σ() = i

|λ

k for every . Since B is analytic, we get that B(σ(), ) =

I for every

∈ R. Now if we go back to the system (E

σ(),

), we can assume,

by continuity and Corollary 2.2, that for the solution (φ, ψ), the function φ is
dominating for every , i.e.

|φ(t, )| > |ψ(t, )|. The winding number j

= Ind(φ) is

then constant and we get from the ﬁrst equation of (E

σ(),

) that

2π

(t, )

φ(t, )

dt = λ

= (σ()

− λ

ν) +

2πi

2π

c(t)

ψ(t, )

φ(t, )

dt.

By taking the limit, we obtain

lim

→0

σ() = lim

→0

|λ

k = lim

→0

+ ν) +

2πi

2π

c(t)

ψ(t, )

φ(t, )

Since the right hand side is bounded and λ

= a + ib with a > 0 and b

= 0, then

necessarily k = 0 and this is a contradiction. This proves that B(σ

)

= I.

The following corollary is a direct consequence of Proposition 2.18 and formula

(2.5).

Corollary

2.19. If 1 is an eigenvalue of the monodromy matrix B(σ, ) with

= 0, then it has multiplicity one.

ABDELHAMID MEZIANI

2.9. Main result about basic solutions

The following theorem summarizes the main properties of the basic solutions

Theorem

2.20. For every j

∈ Z there are exactly two R-independent basic

solutions w

+
j

(r, t, ) and w

−

(r, t, ) of

with Char(w

) = (σ

, j) such that the

spectral values σ

∈ Spec(L

) satisfy

• σ

() depends continuously on and

• if for some

∈ R, σ

(

)

∈ C\R, then σ

−

(

) = σ

(

Proof.

Consider the analytic variety

Γ =

{(σ, ) ∈ C × R; F (σ, ) = 0}

where F is the spectral function given in (2.6). Thus the real spectral values τ

(see Proposition 2.6) satisfy (τ

−

, 0) and (τ

, 0)

∈ Γ. Let Γ

be connected

components of Γ containing (τ

, 0) and Γ

= Γ

−

∪ Γ

+
j

For j

= k we have Γ

∩ Γ

∅. Indeed, if there is (σ

)

∈ Γ

∩ Γ

(with

= 0), then equation (E

) would have periodic solutions,

(t, σ

)

(t, σ

)

and

(t, σ

)

(t, σ

)

giving rise to basic solutions

= r

+ r

and

= r

+ r

with winding numbers j and k, respectively. But the monodromy matrix B(σ

)

has only one eigenvector with eigenvalue 1 (Corollary 2.19). This means φ

= cφ

and ψ

= cψ

for some constant c. Hence, w

= r

cφ

+ r

cψ

has also winding

number j. This is a contradiction.

To complete the proof, we need to show that for every j

∈ Z and for every

∈ R

|Γ

∩ {(σ,

); σ

∈ C}| ≤ 2 ,

where

|S| denotes the cardinality of the set S. By contradiction, suppose that there

exists j

∈ Z and

∈ R

∗

such that

|Γ

∩ {(σ,

); σ

∈ C}| ≥ 3 .

Let (σ

), (σ

), and (σ

) be three distinct points in Γ

. Hence, Γ

has

three distinct components C

, C

, and C

over a neighborhood of

. They are

deﬁned by functions f

(), f

() and f

(), that are analytic, except possibly at

By analytic continuation, Γ

has three distinct branches C

, C

, parametrized

by . That is C

{(f

(), );

∈ R} with f

∈ C

(

R) and analytic everywhere,

except on a set of isolated points. In particular for = 0, we get

(0), f

(0)

} = {τ

−

, τ

In the case τ

−

< τ

, we can assume that f

(0) = f

(0) and this contradicts

Proposition 2.14. In the case τ

−

= τ

, we would have f

(0) = f

(0) = τ

and this would contradict Proposition 2.17.

For the adjoint operator we have the following theorem.

2.9. MAIN RESULT ABOUT BASIC SOLUTIONS

Theorem

2.21. Let

(r, t) = r

σ()

φ(t, ) + r

σ()

ψ(t, )

be a basic solution of

with Char(w

) = (σ(), j). Then

∗

has a basic solution

(r, t) = r

−σ()

X(t, ) + r

−σ()

Z(t, )

with Char(w

) = (

−σ(), −j).

Proof.

For σ()

∈ Spec(B(., )), it follows from (2.17) and (2.16) that σ(−) =

σ() and σ() = σ(

−) ∈ Spec(B(., −)). Let U() be a continuous eigenvector with

eigenvalue 1 of B(σ(

−), −) such that

U (0) =

φ(0, 0)
ψ(0, 0)

Then V(t, σ(

−), −)U() is a periodic solution of (E

σ(

−),−

). The function

X(t, )
Z(t, )

= DV(t, σ(

−), −)U() =

V(t,

−σ(), )DU()

is a periodic solution of the adjoint system (

−σ(),

). Furthermore,

|X| > |Z|

and Ind(X) =

−j so that the character of the associated basic solution is

(

−σ(), −j).

CHAPTER 3

Example

We give here an example in which the basic solutions can be explicitly deter-

mined. This is the case when c(t) = ic

ikt

with c

∈ C

∗

. For simplicity, we assume

that ν = 0. The system of equations (E

σ,

) is

φ(t)

= iσφ(t) + ic

ikt

ψ(t)

−iσψ(t) − ic

−ikt

φ(t).

In this case we can use Fourier series to determine the spectral values and the
periodic solutions.

For a given j

∈ Z, the system has a solution of the form

φ(t) = e

ijt

, ψ(t) = De

i(j

−k)t

with σ and D satisfying

j = σ + c

− k)D = −σD − c

The elimination of D, gives the following quadratic equation for the spectral value
σ

− [(λ

− λ

)j + kλ

]σ

− [j(j − k)|λ

] = 0.

After replacing λ

by a + ib we get σ and D:

= ibj +

− ib)k

−

− ib)k

(a + ib)j

− σ

The corresponding basic solution of

(r, t, ) = r

ijt

+ r

i(j

−k)t

The character of w

is (σ

, j) if

| < 1 and it is (σ

, k

− j) if |D

| > 1.

Note that in order for D

to have norm 1, for some j

, say D

= e

iα

, the

exponent σ

needs to satisfy

= λ

− c

iα

and

−λ

− k) − c

−iα

Consequently, λ

(2j

− k) = σ

− σ

. Since Re(λ

) = a > 0, then necessarily

k = 2j

is an even integer.

Thus for k odd,

| = 1 for every j ∈ Z and for an even k, k = 2j

, we have

| = 1 for every j = j

. At the level j

, we get

= aj

−b

and

ibj

−

−b

and the character of the corresponding basic solution is (σ

, j

CHAPTER 4

Asymptotic behavior of the basic solutions of

In this chapter, we determine the asymptotic behavior of the basic solutions.

This behavior will be needed in the next chapter to estimate the kernels. From now
on, there is no need anymore for the parameter . So we will denote

L and

the associated system of diﬀerential equations (E

σ,1

) by (E

). We will assume here

that λ = a + ib with a > 0 and b

= 0, since the asymptotic behavior in case b = 0

is known from [10]. Hence,

Lu = Lu + iλνu − c(t)u ,

where L is the vector ﬁeld given in (1.1):

L = λ

∂

∂t

− ir

∂

∂r

Let

(4.1)

γ =

4aπ

2π

|c(t)|

and

k(t) =

γt

−

|c(s)|

Note that k(t) is 2π-periodic. We have the following theorem

Theorem

4.1. For j

∈ Z, the operator L has basic solution

(r, t) = r

(t) + r

(t)

with character (σ

, j) such that, as

|j| → ∞, we have

λ(j + ν) +

+ O(j

−2

)

(4.2)

(t)

ijt

1 + i

k(t)

+ O(j

−2

)

(4.3)

(t)

−e

ijt

c(t)

2aj

+ O(j

−2

)

(4.4)

where γ and k(t) are given in (4.1). Furthermore, any basic solution (with

|j| large)

has the form

w(r, t) = r

aφ

(t) + r

aψ

(t),

with a

∈ C.

Remark

4.2. It follows from Theorems 2.21 and 4.1 that for

|j| ∈ Z large, the

basic solutions of

∗

have the form,

∗

(r, t) = r

−σ

−j

(t) + r

−σ

−j

(t)

4. ASYMPTOTIC BEHAVIOR OF THE BASIC SOLUTIONS OF

with Char(w

∗

) = (

−σ

−j) where

−j

(t)

= e

−ijt

− i

k(t)

+ O(j

−2

)

−j

(t)

−e

−ijt

c(t)

2aj

+ O(j

−2

The remainder of this chapter deals with the proof of Theorem 4.1. The proof

will be divided into 3 steps. To simplify the expressions, we will use the following
variables

(4.5)

μ =

− λν

, e

= λ/λ, δ = (e

−ix

+ 1)ν, and c

(t) =

c(t)

The system of equations (E

) becomes then

(4.6)

= iμφ + c

(t)ψ

−ie

(μ + δ)ψ + c

(t)φ.

Now, we proceed with the proof of the theorem.

4.1. Estimate of σ

For a periodic solution (φ, ψ) of (4.6) with

|ψ| < |φ|, we can assume that

max

|φ| = 1 and Ind(φ) = j. Let

(4.7)

T (t) =

ψ(t)

φ(t)

It follows from the ﬁrst equation of (4.6) that

μ +

2πi

2π

(t)T (t)dt =

2πi

2π

φ(t)

dt = j.

Hence,

(4.8)

μ = j

− M

with

2πi

2π

(t)T (t)dt .

Note that

| ≤

2π

(t)

|dt. To obtain a better estimate of M

we use

the second equation of (4.6) to get (after multiplying by c

, dividing by φ and

integrating over [0, 2π])

(4.9)

−ie

(μ + δ)M

−1

2πi

2π

(t)

dt +

2πi

2π

(t)

φ(t)

dt.

We use integration by parts in the last integral together with (4.6) to obtain

2π

(t)

φ(t)

−

2π

(t)T (t)dt +

2π

(t)

(iμφ(t) + c

(t)ψ(t))ψ(t)

φ(t)

2π

(

−c

(t)T (t) + iμc

(t)T (t) + c

(t)

T (t)

)dt.

From this and (4.9), we deduce that

−i

(1 + e

)μ + δe

−1

2πi

2π

(

(t)

+ c

(t)T (t)

− c

2
1

(t)T

(t))dt.

4.1. ESTIMATE OF σ

Consequently,

(4.10)

|(1 + e

)μ + δe

| |M

| ≤ A

where A

2π

(t)

|)dt. Note that since 1 + e

= 2a/λ satisﬁes

Re(1 + e

) > 0 and

|δ| < 2, then it follows from (4.8) and the boundedness of M

that there exists J

∈ Z

such that

(4.11)

|(1 + e

)μ + δe

| ≥

|j|

∀j ∈ Z, |j| ≥ J

Lemma

4.3. Let N

= jM

, then

(4.12)

lim

|j|→∞

−1

2π(1 + e

)

2π

(t)

dt =

−

where γ is given in (4.1).

Proof.

It follows from (4.10) and (4.11) that

| ≤ 2A

. Let

(4.13)

φ(t) = e

ijt

(t)

and

ψ(t) = e

ijt

(t).

Hence, Ind(φ

) = 0 and T = ψ

/φ

. From (4.6), we get the system for φ

, ψ

(4.14)

−i

+ c

(t)ψ

−iE

+ c

(t)φ

where

(4.15)

= (1 + e

− e

+ e

δ .

Note that since

| ≤ 2A

and e

= −1, then for |j| large (|j| ≥ J

), we have

(4.16)

| ≥ |1 + e

|j|

Now, it follows from the ﬁrst equation of (4.14) and from Ind(φ

) = 0 that

(4.17)

2πi

2π

(t)T (t)dt .

We use the second equation of (4.14) to estimate the integral appearing in (4.17)

(4.18)

2π

T dt =

−1

2π

− c

dt =

2π

− c

dt.

We use integration by parts and system (4.14) to evaluate the last integral appearing
in (4.18).

(4.19)

2π

−

2π

T + c

(

−iN

/j)φ

+ c

2π

(

−c

T + c

2
1

)dt

−

2π

T dt.

Thus,

(4.20)

2π

T dt =

2π

(

+ c

− c

2
1

)dt +

2π

T dt.

4. ASYMPTOTIC BEHAVIOR OF THE BASIC SOLUTIONS OF

Therefore, from (4.20) and (4.16), we have

(4.21)

2π

T dt =

2π

dt + I

− I

+ O(

)

where I

and I

are given by

2π

(t)T (t)dt

and

2π

2
1

(t)T

(t)dt.

Now we show that I

= O(1/j) and I

= O(1/j). For I

, we have, after using

system (4.14), that

−1

2π

− c

−1

2π

dt +

2π

T dt

−

2π

−1

2π

(

+ c

T )dt

−

2π

(

−iN

/j)φ

+ c

(4.22)

2π

(

+ c

− c

)dt +

2π

T dt.

Since

|T | < 1 and |N

| < 2A

, we get I

= O(1/j). For I

, we use system (4.14)

and integration by parts, to obtain

(4.23)

−iE

2π

2
1

(ψ

− c

)dt =

−

2π

2
1

T dt +

2π

2
1

(ψ

)

−

2π

2
1

T dt

−

2π

dt +

2π

2
1

−

2π

2
1

T + c

)dt +

2π

2
1

(

−iN

/j)φ

+ c

−

2π

2
1

T + c

− c

3
1

)dt

−

2π

2
1

and again

| is bounded and therefore I

= O(1/j). With these estimates for

and I

, expressions (4.17), (4.20), and (4.21) give

−

2πE

2π

dt + O(

|j|

−1

)

Since lim

|j|→∞

1 + e

, the lemma follows.

Using this lemma we get

−λ

2πj(λ + λ)

2π

|c(t)|

|λ|

dt + O(j

−2

) =

−

λj

+ O(j

−2

Consequently, μ = j

− M

= j +

jλ

+ O(j

−2

) and σ = λ(μ + ν) gives estimate (4.2)

of the theorem.

4.2. FIRST ESTIMATE OF φ AND ψ

4.2. First estimate of φ and ψ

We begin with the estimates of the components φ and ψ of the basic solutions.

Let φ

, and ψ

be the functions deﬁned in (4.13). Note that max

|φ

| ≤ 1 and

|ψ

| < |φ

|. We have the following lemma that gives an estimate of ψ

Lemma

4.4. There exist J

∈ Z

and K > 0 such that the function ψ

= jψ

satisﬁes

(4.24)

|ψ

(t)

| = |jψ

(t)

| ≤ K,

∀t ∈ R, ∀j ∈ Z, |j| ≥ J

Proof.

The system (4.14) implies that T = ψ/φ = ψ

/φ

satisﬁes the equation

(4.25)

(t) =

−L

T (t)

− c

(t)T

(t) + c

(t)

where

(4.26)

= i

−

= i

(1 + e

)j + e

δ + O(1/j)

Note that the real part, p, of L

is given by

p = Re(L

) =

−j sin x + Re(δe

) + O(1/j),

and, since sin x

= 0 (because b = 0), there exists J

∈ Z

such that

(4.27)

|p| ≥

| sin x|

|j|

∀j ∈ Z, |j| ≥ J

Let ρ(t) =

|T (t)|, A(t) = arg T (t), ϑ(t) = arg c

(t) and M = max

≤t≤2π

(t)

|. Let us

rewrite equation (4.25) as

(4.28)

+ iA

ρ =

−L

− |c

|ρ

i(A+ϑ)

−i(A+ϑ)

By taking the real part, we obtain

(4.29)

−pρ − |c

|ρ

cos(A + ϑ) +

| cos(A + ϑ).

Since 0

≤ ρ < 1, we get −2M ≤ ρ

+ pρ

≤ 2M. Equivalently,

(4.30)

−2Me

≤ (ρ(t)e

)

≤ 2Me

In the case where p > 0 (and so p >

|j sin x|/2), we obtain, after integrating (4.30)

from 0 to t, with t > 0, that

(4.31)

ρ(t)

≤

ρ(0)

−

−pt

Let t

> 0 be such that e

−pt

≤ 2M/p. Then, it follows from (4.31), that

(4.32)

≤ ρ(t) ≤

1 + ρ(0)

−

≤

∀t ≥ t

where K is a constant independent on j. Since the function ρ is periodic, then
inequality (4.32) holds for every t

∈ R. When p < 0, an analogous argument

(integrating (4.30) from t to 0 with t < 0) yields the estimate (4.32). Hence,
|T | < K/j and this completes the proof of the lemma.

4. ASYMPTOTIC BEHAVIOR OF THE BASIC SOLUTIONS OF

Throughout the remainder of this chapter, we will use Fourier series. For a

function F

∈ L

(

C), we denote by F

(l)

its l-th Fourier coeﬃcient:

(l)

2πi

2π

F (t)e

−ilt

dt.

An estimate of φ

is given in the following lemma.

Lemma

4.5. There exist J

∈ Z

and positive constants K

, K

such that for

∈ Z, |j| > J

, the function φ

has the form

(4.33)

(t) = φ

(0)
1

(t)

with

|φ

(t)

| ≤ K

for every t

∈ R, φ

(0)
2

= 0, and

≤ |φ

(0)
1

| ≤ K

Proof.

We estimate the Fourier coeﬃcients of φ

in terms of those of the

diﬀerentiable function c

, where ψ

is the function in Lemma 4.4. For this, we

replace ψ

by ψ

/j in (4.14) and rewrite the ﬁrst equation as

(4.34)

(t) =

−i

(t) + c

(t)

In terms of the Fourier coeﬃcients, we get then

(4.35)

l +

(l)
1

)

(l)

∈ Z.

Since N

∼ −

(by Lemma 4.3), then there exists J

∈ Z

such that

l +

≥

|l|

∀l ∈ Z

∗

It follows from (4.35) that

(4.36)

|φ

(l)
1

| ≤

|j|

|(c

)

(l)

|l|

∀l ∈ Z

∗

The function φ

(t) = j(φ

(t)

− φ

(0)
1

) satisﬁes the conditions of the lemma. For

l = 0, we get

(0)
1

−i

)

(0)

and since ψ

is bounded, then

|φ

(0)
1

| ≤ K

. Finally,

|φ

(0)
1

| ≥ 1/2 for |j| large since

|φ

| ≤ K

and max

|φ

| = 1.

4.3. End of the proof of Theorem 4.1

Now we estimate the functions ψ

and φ

that are deﬁned in the previous

lemmas.

Lemma

4.6. There exist J

∈ Z

and K > 0 such that for j

∈ Z, |j| ≥ J

, the

function ψ

, given in Lemma 4.4, has the form

(4.37)

(t) = φ

(0)
1

(t)

i(1 + e

)

(t)

with ψ

satisfying

|ψ

(t)

| ≤ K for every t ∈ R.

4.3. END OF THE PROOF OF THEOREM 4.1

Proof.

With φ

and ψ

as in Lemmas 4.4 and 4.5, we rewrite the system

(4.14) and obtain

(4.38)

(t)

−iN

(0)
1

(t)

+ c

(t)ψ

(t)

−iE

(t) + c

(t)(jφ

(0)
1

+ φ

(t)).

From (4.38), we see that the Fourier coeﬃcients satisfy

(4.39)

(l)

= φ

(0)
1

i(l + E

)

(l)

)

(l)

i(l + E

)

∈ Z.

Let

(4.40)

(t) = ψ

(t)

− φ

(0)
1

(t)

i(1 + e

)

It follows at once from (4.39) that the Fourier coeﬃcients of ψ

satisfy

(4.41)

(l)

= φ

(0)
1

(1 + e

− (l + E

)

i(l + E

)(1 + e

)

(l)

)

(l)

i(l + E

)

∈ Z.

Since for

|j| large, we have

|(1 + e

− E

| = |

− δ| < 3 and |l + E

| ≥ |Im(E

)

| ≥

for some positive constant C

, then from (4.41) we get

(4.42)

|ψ

(l)

| ≤

|φ

(0)
1

(l)

| + |(c

)

(l)

The term ilc

(l)

is the l-th Fourier coeﬃcient of c

∈ C

−1

, and therefore it follows

from (4.42) that jψ

is uniformly bounded (in j). Thus the function ψ

∈

−1

is bounded.

Lemma

4.7. There exist J

∈ Z

and K > 0 such that for j

∈ Z, |j| ≥ J

, the

function φ

, given in Lemma 4.5, has the form

(4.43)

(t) = iφ

(0)
1

k(t) +

(t)

where k(t) is the function given in (4.1), and φ

∈ C

satisﬁes

|φ

(t)

| ≤ K for

every t

∈ R.

Proof.

We use Lemma 4.3 to write

−

with C

∈ C bounded. By using Lemmas 4.5 and 4.6 in (4.34), we obtain an

equation for φ

(4.44)

(t) = (i

−

)(φ

(0)
1

(t)

) +

(0)
1

(t)

i(1 + e

)

(t)ψ

(t)

4. ASYMPTOTIC BEHAVIOR OF THE BASIC SOLUTIONS OF

Equivalently,

(4.45)

(t) = iφ

(0)
1

−

(t)

1 + e

jφ

(0)
1

−

)φ

(t) + c

(t)ψ

(t)

Let φ

(t) = φ

(t)

− iφ

(0)
1

k(t), where k(t) is the function deﬁned in (4.1). Note that

since k

(t) =

−

(t)

1 + e

, then it follows from (4.45) that

(4.46)

(t) =

−iC

+ c

(t)ψ

(t) + i

−

(φ

(t) + iφ

(0)
1

k(t))

The Fourier coeﬃcients of φ

satisfy

(4.47)

−

λj

(l)
3

−iC

+ (c

)

(l)

−

(0)
1

(l)

This shows that jφ

is bounded.

From these lemmas, we get for the functions φ and ψ

φ(t)

= e

ijt

0
1

+ iφ

0
1

k(t)

(t)

ψ(t)

= e

ijt

0
1

(t)

ij(1 + e

)

(t)

Since

≤ φ

(0)
1

≤ K

, then we can divide the above functions by φ

(0)
1

and obtain

(4.3) and (4.4) of the Theorem.

CHAPTER 5

The kernels

We use the basic solutions to construct kernels for the operator

L. For j, k ∈ Z,

let

(5.1)

(r, t)

= r

±
j

(t) + r

±
j

(t)

∗±

(r, t)

= r

±
k

(t) + r

±
k

(t)

be the basic solutions of

L and L

∗

, respectively, with Char(w

) = (σ

, j), Char(w

∗±

)

= (μ

, k) and such that φ

+
j

(0) = 1, φ

−

(0) = i, X

(0) = i, and X

−

(0) = 1. Note

that, by Theorem 2.21, we have μ

−σ

−j

In the remainder of this paper we will use the following notation,

= A

+ A

−

Functions f (r, t) and g(ρ, θ) will be denoted by f (z) and g(ζ), where z = r

and

ζ = ρ

iθ

. Deﬁne functions Ω

(z, ζ) and Ω

(z, ζ) as follows:

(5.2)

(z, ζ) =

⎧

⎪

⎨
⎪

⎪

⎩

Re(σ

±
j

)

≥0

(z)w

∗±

−j

(ζ)

if r < ρ

−

Re(σ

±
j

)<0

(z)w

∗±

−j

(ζ)

if r > ρ

and

(5.3)

(z, ζ) =

⎧

⎪

⎨
⎪

⎪

⎩

Re(σ

±
j

)

≥0

(z)w

∗±

−j

(ζ)

if r < ρ

−

Re(σ

±
j

)<0

(z)w

∗±

−j

(ζ)

if r > ρ.

Let K(t, θ) and L(z, ζ) be deﬁned by

(5.4)

K(t, θ) = k(t)

− k(θ) and L(z, ζ) =

⎧

⎪

⎨
⎪

⎩

log

− z

if r < ρ

log

− ζ

if r > ρ

where k(t) is the function deﬁned in (4.1) and where log denotes the principal
branch of the logarithm in

C\R

−

. In the next theorem, we will use the notation

{(r, t, ρ, θ); 0 < r ≤ ρ}, Δ

{(r, t, ρ, θ); 0 < ρ ≤ r}

and Int(Δ

), Int(Δ

) will denote their interiors.

5. THE KERNELS

Theorem

5.1. The functions C

(z, ζ) and C

(z, ζ) deﬁned in Int(Δ

)

∪Int(Δ

)

(5.5)

(z, ζ)

= Ω

(z, ζ)

− i

λν

− z

+ iK(t, θ)L(z, ζ)

(z, ζ)

= Ω

(z, ζ)

−

c(t)

λν

L(z, ζ)

−

c(θ)

λν

L(z, ζ)

are in C

(Δ

)

∪ C

(Δ

), meaning that the restrictions of C

1,2

to Int(Δ

) (or to

Int(Δ

)) extend as C

functions to Δ

(or Δ

). Furthermore, for any R > 0, the

functions C

1,2

are bounded for r

≤ R and ρ ≤ R.

To prove this theorem, we need two lemmas.

5.1. Two lemmas

Lemma

5.2. For

|j| large, we have

(5.6)

(z)w

∗±

−j

(ζ)

= 2i

ij(t

−θ)

1 + i

K(t, θ)

+ O(j

−2

)

(z)w

∗±

−j

(ζ)

c(t)

ij(t

−θ)

c(θ)

−ij(t−θ)

+ O(j

−2

Proof.

It follows from Theorem 4.1 that for large

|j|, the corresponding spec-

tral values are in

C\R (for b = 0). We have, σ

−

= σ

and σ

is given by

(4.2). Furthermore, it follows from Chapter 2 that

+
j

(r, t) = r

(t) + r

(t)

and

−

(r, t) = i(r

(t)

− r

(t))

with φ

(0) = 1. For the basic solutions of the adjoint operator, we have (from

Theorem 2.21) that

∗+

−j

(ρ, θ) = i(ρ

−σ

−j

(θ)

− ρ

−σ

−j

(θ))

and

∗−

−j

(ρ, θ) = ρ

−σ

−j

(θ) + ρ

−σ

−j

(θ)

with X

−j

(0) = 1. Hence

(5.7)

(z)w

∗±

−j

(ζ)

= 2i

(t)X

−j

(θ)

−

(t)Z

−j

(θ)

(z)w

∗±

−j

(ζ)

= 2i

(t)X

−j

(θ)

−

(t)Z

−j

(θ)

Now, the asymptotic expansions (4.3) and (4.4) give the following products

(5.8)

(t)X

−j

(θ)

= e

ij(t

−θ)

1 + i

K(t, θ)

+ O(j

−2

)

(t)X

−j

(θ)

−i

c(t)

2aj

ij(t

−θ)

+ O(j

−2

)

(t)Z

−j

(θ)

−i

c(θ)

2aj

ij(t

−θ)

+ O(j

−2

)

(t)Z

−j

(θ)

= O(j

−2

Estimates (5.6) of the lemma follow from (5.7) and (5.8).

5.1. TWO LEMMAS

Lemma

5.3. For j

∈ Z

large and σ

as in (4.2), consider the function

(t) = t

− t

λ(j+ν)

0 < t < 1 .

Then there are J

> 0 and C > 0 such that

(t)

| ≤

∀t ∈ (0, 1), j ≥ J

Proof.

By using the asymptotic expansion for σ

given in (4.2) and λ = a + ib

(a > 0), we write

a(j + ν) +

+ i

b(j + ν) +

2β

where α > 0 and β

∈ R, depend on j, but are bounded. Hence,

(t) = t

a(j+ν)+(α/j)

i[b(j+ν)+2(β/j

)]

− t

a(j+ν)

ib(j+ν)

We decompose f

as f

= g

+ h

with

(t)

= t

a(j+ν)

i(b(j+ν)+2(β/j

))

α/j

− 1

(t)

= t

a(j+ν)

ib(j+ν)

2iβ/j

− 1

It is enough to verify that both

|g| and |h| are O(1/j

). Since a > 0 and j large,

then we can assume that g and h are deﬁned at 0 and that

g(0) = h(0) = g(1) = h(1) = 0 .

For the function g, we have

|g(t)| = t

a(j+ν)

− t

a(j+ν)+(α/j)

The maximum of

|g| occurs at the point

∗

a(j + ν)

a(j + ν) + (α/j)

j/α

and

|g(t

∗

)

| = t

a(j+ν)

∗

−

a(j + ν)

a(j + ν) + (α/j)

≤

j(a(j + ν) + (α/j))

≤

for some positive constant A

For the function h, note that if β = 0, then h = 0. So assume that β

= 0. We

have

|h(t)|

= t

2a(j+ν)

iβ/j

− t

−iβ/j

= 4t

2a(j+ν)

sin

β ln t

For 0 < t

≤ 1/2, we get

|h(t)| ≤ 2t

a(j+ν)

≤

a(j+ν)

≤

for some A

> 0. For 1

≥ t ≥ (1/2), we have

(

|h(t)|

) = 8t

a(j+ν)

−1

sin(

β ln t

)

a(j + ν) sin(

β ln t

) +

cos(

β ln t

)

5. THE KERNELS

For j suﬃciently large, the critical points of

|h|

in the interval (1/2 , 1) are the

solutions of the equation

tan

β ln t

−

(j + ν)

Hence,

β ln t

− arctan(

(j + ν)

) + kπ,

∈ Z.

However, since 1

≥ t ≥ 1/2 and j is large, the only possible value of the integer k

is k = 0. Hence,

|h| has a single critical point in (1/2, 1):

∗

= exp

−

arctan(

(j + ν)

)

The function

|h| has a maximum at t

∗

and

|h(t

∗

)

= 4t

2a(j+ν)

∗

sin

(arctan(

(j + ν)

))

≤ 4 arctan

(

(j + ν)

)

≤

for some A

> 0.

5.2. Proof of Theorem 5.1

We use the series expansions

− z

⎧

⎪

⎨
⎪

⎪

⎩

≥0

(r/ρ)

λj

ij(t

−θ)

if r < ρ

−

≥1

(r/ρ)

−λj

−ij(t−θ)

if r < ρ

L(z, ζ)

⎧

⎪

⎨
⎪

⎪

⎩

≥1

(r/ρ)

λj

ij(t

−θ)

if r < ρ

≥1

(r/ρ)

−λj

−ij(t−θ)

if r < ρ

together with (5.2) and (5.6) to decompose C

(z, ζ) as follows. For a large integer

and r < ρ,

(5.9)

= P

+ i

≥J

−

λ(j+ν)

ij(t

−θ)

1 + i

+ O(j

−2

)

where P

(z, ζ) consists of the ﬁnite collection of terms in the series with index

j < J

. Thus P

∈ C

(Δ

). The second term (

≥J

· · · ) on the right of (5.9) is

also in C

(Δ

) since

−

λ(j+ν)

= O(j

−2

)

by Lemma 5.3. When r > ρ, the decomposition of C

takes the form

(5.10) C

≥J

−λ(j−ν)

−

−j

−ij(t−θ)

− i

+ O(1/j

)

5.3. MODIFIED KERNELS

As before, the ﬁnite sum

(z, ζ)

∈ C

(Δ

). Since

−j

= λ(

−j + ν) − (γ/j) + O(j

−2

)

and r > ρ, we have

−λ(j−ν)

−

−j

λ(j

−ν)

−

−σ

−j

= O(j

−2

Again, the inﬁnite sum on the right of (5.10) is in C

(Δ

). This proves the theorem

for the function C

. Similar arguments can be used for the function C

5.3. Modiﬁed kernels

The following modiﬁcations to the kernels Ω

and Ω

will be used to establish

a similarity principle in Chapter 8. For j

∈ Z, we deﬁne Ω

(z, ζ) and Ω

(z, ζ)

(5.11)

(z, ζ) =

⎧

⎪

⎨
⎪

⎪

⎩

Re(σ

±
j

)

≥Re(σ

±
j0

)

(z)w

∗±

−j

(ζ)

if r < ρ

−

Re(σ

±
j

)<Re(σ

±
j0

)

(z)w

∗±

−j

(ζ)

if r > ρ

and

(5.12)

(z, ζ) =

⎧

⎪

⎨
⎪

⎪

⎩

Re(σ

±
j

)

≥Re(σ

±
j0

)

(z)w

∗±

−j

(ζ)

if r < ρ

−

Re(σ

±
j

)<Re(σ

±
j0

)

(z)w

∗±

−j

(ζ)

if r > ρ.

Theorem

5.4. The functions C

(z, ζ) and C

(z, ζ) given by

(5.13)

(z, ζ) = i

±
j0

−θ)

− z

+ iK(t, θ)L(z, ζ) + C

(z, ζ)

and

(5.14)

(z, ζ)

c(t)

±
j0

L(z, ζ) +

c(θ)

±
j0

L(z, ζ)

±
j0

(z, ζ)

have the following properties:

(1) C

1,2

∈ C

(Int(Δ

)

∪ Int(Δ

)),

(2) for a given z = r

with 0 < r < R, the functions C

1,2

(z,

·) are in

(

{(ρ, θ); ρ < R}), for every p > 0, and

(3) the functions

|z − ζ|

1,2

are bounded in the region r < R and ρ < R, for

any > 0.

5. THE KERNELS

Proof.

The proof follows similar arguments as those used in the proof of

Theorem 5.1. We describe brieﬂy how the properties of C

can be established in

the region r < ρ. We write

− i

±
j0

−θ)

− z

+ iK(t, θ)L(z, ζ)

= P

(z, ζ) +

≥J

where P

is the ﬁnite sum consisting of terms in the series of Ω

with indices

≤ J

and Re(σ

)

≥ Re(σ

), and the terms with indices j

≤ J

in the series

expansions of ζ/(ζ

− z) and L(z, ζ). The inﬁnite sum

≥J

can be written as

≥J

= i

±
j0

≥J

(z, ζ)

where

(z, ζ) =

−σ

±
j0

−

λ(j

−j

)

ij(t

−θ)

1 + i

K(t, θ)

+ O(j

−2

)

Since σ

satisﬁes the asymptotic expansion (4.2) and since r < ρ, arguments similar

to those used in the proof of Lemma 5.3 show that

−σ

±
j0

−

λ(j

−j

)

= O

Thus

≥J

has the desired properties of the theorem in Δ

. Analogous arguments

can be used in the region r > ρ and also for the function C

CHAPTER 6

The homogeneous equation

Lu = 0

In this chapter, we use the kernels deﬁned in Chapter 5 to obtain series and

integral representations of the solutions of the equation

Lu = 0. Versions of the

Laurent series expansion, in terms of the basic solutions, and the Cauchy integral
formula are derived. Some consequences of these representations are given.

6.1. Representation of solutions in a cylinder

For R, δ

∈ R

with δ < R, consider the cylinder A(δ, R) = (δ, R)

× S

. Again,

let z = r

, ζ = ρ

iθ

and Ω

, Ω

denote the functions deﬁned in Chapter 5. We

have the following theorem.

Theorem

6.1. Let u

∈ C

(A(δ, R)) be a solution of

Lu = 0. Then

(6.1)

u(z) =

−1

2π

∂A(δ,R)

(z, ζ)

u(ζ)dζ +

(z, ζ)

u(ζ) dζ.

Proof.

Let

L = λ

∂

∂t

− ir

∂

∂r

and

∗

= λ

∂

∂θ

− iρ

∂

∂ρ

so that

Lu = Lu + iλνu − c(t)u

and

−L

∗

v = L

∗

− iλνv + c(θ)v .

It follows from the deﬁnitions of the kernels Ω

1,2

given in (5.2) and (5.3) and from

the fact that

= 0 and

∗

∗±

= 0 that the kernels satisfy the following relations

(6.2)

∗

(z, ζ) = iλνΩ

(z, ζ)

− c(θ) Ω

(z, ζ)

∗

(z, ζ) = iλνΩ

(z, ζ)

− c(θ) Ω

(z, ζ).

Consider the functions

= Ω

+ Ω

and

−iΩ

+ iΩ

Then (6.2) implies that

(6.3)

∗

= 0 .

Let (r

, t

)

∈ A(δ, R) and z

= r

. For > 0, let

{(ρ, θ) ∈ R

× S

|ζ − z

| < } .

6. THE HOMOGENEOUS EQUATION

Lu = 0

Hence, for small, D

is diﬀeomorphic to the disc and is contained in A(δ, R). We

apply Green’s identity (1.8) in the region A(δ, R)

to each pair u(ζ), P

, ζ),

with k = 1, 2. Since,

Lu = 0 and L

∗

= 0, then

∂A

, ζ)u(ζ)

dζ

+ P

, ζ) u(ζ)

dζ

∂D

, ζ)u(ζ)

dζ

+ P

, ζ) u(ζ)

dζ

Then, after multiplying by i the above identity with k = 2 and adding it to the
identity with k = 1, we obtain

∂A

+ iP

dζ

+ (P

+ iP

dζ

∂D

+ iP

dζ

+ (P

+ iP

dζ

Since 2Ω

= P

+ iP

and 2Ω

= P

− iP

, we get

(6.4)

∂A

, ζ)u(ζ)

dζ

+ Ω

, ζ) u(ζ)

dζ

∂D

, ζ)u(ζ)

dζ

+ Ω

, ζ) u(ζ)

dζ

Now, we let

→ 0 in the right side of (6.4). From the estimates (5.5) of the kernels,

it follows that the only term that provides a nonzero contribution (as

→ 0) is the

term containing ζ/(ζ

− z) since C

, C

are bounded and L(z

, ζ) has a logarithmic

growth. That is,

lim

→0

∂D

dζ

+ Ω

dζ

= lim

→0

∂D

i(r

/ρ)

λν

u(ζ)

− z

dζ =

−2πu(z

This proves the Theorem.

Theorem

6.2. Suppose that u is a solution of

Lu = 0 in the cylinder A(δ, R)

with 0

≤ δ ≤ R ≤ ∞. Then, u has the Laurent series expansion

(6.5)

u(r, t) =

∈Z

(r, t)

where a

∈ R are given by

(6.6)

−1

2π

∗±

−j

, θ)u(R

, θ)idθ

where R

is any point in (δ, R). Furthermore, there exists C > 0 such that

(6.7)

| ≤

Re(σ

±
j

)

max

|u(R

, θ)

∀j ∈ Z.

Proof.

Let R

∈ (δ, R), and δ

, R

be such that

δ < δ

< R

< R .

For r

∈ (δ

, R

), we apply the integral representation (6.1) in the cylinder A(δ

, R

)

to get

(6.8)

−2πu(r, t) = I

− I

6.1. REPRESENTATION OF SOLUTIONS IN A CYLINDER

where

ρ=R

(z, ζ)u(ζ)

dζ

+ Ω

(z, ζ) u(ζ)

dζ

ρ=δ

(z, ζ)u(ζ)

dζ

+ Ω

(z, ζ) u(ζ)

dζ

The series (5.2) and (5.3) for Ω

and Ω

give

Re(σ

±
j

)

≥0

(r, t)Re

ρ=R

∗±

−j

(ζ)u(ζ)

dζ

Since

Lu = 0 and L

∗

∗±

= 0, then Green’s identity gives

ρ=R

∗±

−j

(ζ)u(ζ)

dζ

= Re

2π

∗±

−j

, θ)u(R

, θ)idθ .

Hence,

−2π

Re(σ

±
j

)

≥0

(r, t)

where a

is given by (6.6). A similar calculation shows that

= 2π

Re(σ

±
j

)<0

(r, t) .

To estimate the coeﬃcients a

, recall that

∗±

−j

, θ) = R

−σ

±
j

−j

(θ) + R

−σ

±
j

−j

(θ) .

Thus

∗±

−j

, θ)

| ≤

Re(σ

±
j

)

−j

(θ)

| + |Z

−j

(θ)

≤

Re(σ

±
j

)

where C = sup

k,θ

(θ)

| + |Z

(θ)

. This gives estimate (6.7).

The following theorem is a direct consequence of Theorem 6.2.

Theorem

6.3. Let u be a bounded solution of

Lu = 0 in the cylinder A(0, R).

Then u has the series expansion

(6.9)

u(r, t) =

Re(σ

±
j

)

≥0

(r, t)

where a

are given by (6.6). If, in addition, u is continuous on A(0, R), then the

above summation is taken over the spectral values σ

satisfying Re(σ

) > 0 or

= 0.

6. THE HOMOGENEOUS EQUATION

Lu = 0

6.2. Cauchy integral formula

For a subset U

⊂ R × S

, we set ∂

U = ∂U

, where S

{0} × S

. We have

the following integral representation that generalizes the classical Cauchy integral
formula.

Theorem

6.4. Let U be an open and bounded subset of

× S

such that

∂U consists of ﬁnitely many simple closed and piecewise smooth curves. Let u

∈

) be such that

Lu = 0. Then, for (r, t) ∈ U, we have

(6.10)

u(r, t) =

−1

2π

∂

(z, ζ)u(ζ)

dζ

+ Ω

(z, ζ) u(ζ)

dζ

Proof.

For δ > 0, deﬁne U

= U

\A(0, δ). Let (r

, t

)

∈ U. Choose > 0 and

δ > 0 small enough so that (r

, t

)

∈ U

and D

⊂ U

, where

{(ρ, θ); |ζ − z

| < } .

Arguments similar to those used in the proof of Theorem 6.1 show that

(6.11)

u(r

, t

) =

−1

2π

∂U

(z, ζ)u(ζ)

dζ

+ Ω

(z, ζ) u(ζ)

dζ

If S

∩ ∂U = ∅, then for δ small enough U

= U and the theorem is proved in this

case. If S

∩ ∂U = ∅, let

= ∂U

∩

{δ} × S

That is, Γ

is the part of ∂U

contained in the circle r = δ. Going back to the

deﬁnition of the kernels, we obtain
(6.12)

(z, ζ)u(ζ)

dζ

+ Ω

(z, ζ) u(ζ)

dζ

−

Re(σ

±
j

)<0

, t

)Re

∗±

−j

(ζ)u(ζ)

dζ

Since there exists C > 0 such that

∗±

−j

(ρ, θ)

| ≤ Cρ

−Re(σ

±
j

)

∀j ∈ Z

then, it follows from (6.12), that

(z, ζ)u(ζ)

dζ

+ Ω

(z, ζ) u(ζ)

dζ

≤ 2πC||u||

Re(σ

±
j

)<0

−Re(σ

±
j

)

, t

)

Therefore,

lim

→0

(z, ζ)u(ζ)

dζ

+ Ω

(z, ζ) u(ζ)

dζ

= 0

and (6.10) follows from (6.11) when we let δ

→ 0.

The following theorem extends the Cauchy integral formula to include the

points on the characteristic circle S

, when the solution is continuous up to the

boundary.

6.3. CONSEQUENCES

Theorem

6.5. Suppose that

L has no spectral values in iR

∗

. Let U be an open,

bounded subset of

×S

, such that ∂U consists of ﬁnitely many simple closed and

piecewise smooth curves and with S

⊂ ∂U. Let u ∈ C

(U ) be such that

Lu = 0.

Then, for (r, t)

∈ U ∪ S

, we have

(6.13)

u(r, t) =

−1

2π

∂

(z, ζ)u(ζ)

dζ

+ Ω

(z, ζ) u(ζ)

dζ

Proof.

We know from Theorem 6.4 that (6.13) holds for r > 0. We need to

verify that it holds at the points (0, t)

∈ S

. Since

L has no purely imaginary spec-

tral values, then Ω

and Ω

are well deﬁned on S

. We have Ω

(0, t) = Ω

(0, t) = 0

if 0 is not a spectral value of

L and if 0 is a spectral value, with say multiplicity 2,

then

(0, t, ρ, θ)

= f

(t)g

−j

(θ) + f

−

(t)g

−

−j

(θ)

(0, t, ρ, θ)

= f

(t)g

−j

(θ) + f

−

(t)g

−

−j

(θ)

where f

(t) are the basic solutions of

L with exponent 0 and g

−j

(t) are the basic

solutions of

∗

with exponent 0.

When 0 is not a spectral value, (6.13) holds for r = 0 by letting r

→ 0 in (6.10).

In this case u

≡ 0 on S

. When 0 is spectral value, then since S

⊂ ∂U, u(r, t) has

a Laurent series expansion in a cylinder A(0, δ)

⊂ U. In particular,

u(0, t)

= a

(t) + a

−

(t)

−1

2π

∂

(0, t, ζ)u(ζ)

dζ

+ Ω

(0, t, ζ) u(ζ)

dζ

Remark

6.6. It follows from this Theorem that if

L has no spectral values on

∗

, then all bounded solutions of

Lu = 0 in a cylinder A(0, R) are continuous up

to S

. If

L has spectral values on iR

∗

, then there are bounded solutions on A(0, R)

that are not continuous up to S

. In fact, the basic solution r

iτ

φ(t) + r

−iτ

ψ(t) is

such a solution when iτ

∈ Spec(L). Note also that the number of spectral values

on i

∗

is at most ﬁnite (this follows from the asymptotic expansion of σ

6.3. Consequences

We give here some consequences of the above representation theorems. First

we deﬁne the order of a solution along S

. We say that a solution, u, of

Lu = 0 in

a cylinder A(0, R) has a zero or a pole of order s = Re(σ

) (with σ

= σ

or σ

−

)

along the circle S

if, in the Laurent series expansion of u, all the coeﬃcients a

corresponding to Re(σ

) < s are zero. That is

u(r, t) =

Re(σ

±
j

)

≥s

(r, t) .

We have the following uniqueness result.

Theorem

6.7. Suppose that the spectral values of

L satisfy the following con-

dition

(6.14)

Re(σ

) = Re(σ

) =

⇒ σ

= σ

∀σ

, σ

∈ Spec(L) .

6. THE HOMOGENEOUS EQUATION

Lu = 0

Let u be a solution of

Lu = 0 in a cylinder A(0, R). Suppose that u is of ﬁnite

order along S

and that there is a sequence of points (r

, t

)

∈ A(0, R) such that

→ 0 and u(r

, t

) = 0 for every k

∈ Z

. Then u

≡ 0.

Proof.

By contradiction, suppose that u

≡ 0. Let s be the order of u on S

First, consider the case where s

∈ R is a spectral value (say of multiplicity 2). Let

(t) and r

−

(t) be the corresponding basic solutions. The function u has the

form

u(r, t) = r

−

(t) + a

(t)) + o(r

The functions f

and f

−

are independent solutions of the ﬁrst order diﬀerential

equation (2.3) and a

∈ R (not both zero). We can assume t

→ t

as k

→ ∞. It

follows from the above representation of u and from the hypothesis u(r

, t

) = 0

that lim

→∞

(u(r

, t

)/r

) = 0. Consequently,

−

f (t

) + a

) = 0.

Thus, the solution a

−

+ a

of (2.3) is identically zero (by uniqueness). Hence,

−

= a

= 0 which is a contradiction.

If s is not a spectral value, then (by condition (6.14)), it must be the real part of

a unique spectral value σ

∈ C\R. The corresponding R-independent basic solutions

are

s+iβ

φ(t) + r

s+iβ

ψ(t)

and

i(r

s+iβ

φ(t)

− r

s+iβ

ψ(t))

with β

∈ R

∗

and

|φ(t)| > |ψ(t)| for every t ∈ R. The Laurent series of u starts as a

linear combination of these two basic solution and u can then be written as

u(r, t) = r

+ ia

−

iβ

φ(t) + (a

− ia

−

iβ

ψ(t)

+ r

Φ(r, t)

with a

∈ R (not both zero), τ > s and Φ a bounded function. It follows from the

assumption u(r

, t

) = 0 that for every k

∈ Z, we have

+ ia

−

)φ(t

) + (a

− ia

−

2iβ

ψ(t

) + r

−s−iβ

Φ(r

, t

) = 0.

But this is only possible when a

+ ia

−

= 0 (since

|φ| > |ψ|, r

→ 0 and Φ

bounded).

The next theorem deals with sequences of solutions that converge on the dis-

tinguished boundary ∂

U = ∂U

Theorem

6.8. Let U be an open and bounded subset of

×S

whose boundary

consists of ﬁnitely many simple, closed and piecewise smooth curves. Let u

(r, t) be

a sequence of bounded functions with u

∈ C

) such that

= 0 for every

n. If u

converges uniformly on ∂

U = ∂U

, then u

converges uniformly on

to a solution u of

Lu = 0.

Proof.

For (r, t)

∈ ∂

U , let Φ(r, t) = lim

→∞

(r, t). The function u(r, t)

deﬁned in U by

u(r, t) =

−1

2π

∂

(r, t, ζ)Φ(ζ)

dζ

+ Ω

(r, t, ζ) Φ(ζ)

dζ

solves

Lu = 0 (since for each ﬁxed ζ, LΩ

(z, ζ) =

L Ω

(z, ζ) = 0). Now, the Cauchy

integral formula applied to u

, shows that u is the uniform limit of u

inside U .

6.3. CONSEQUENCES

The following Liouville property is a direct consequence of the Laurent series

expansion and estimate (6.7) of the coeﬃcients.

Theorem

6.9. Let u be a bounded solution of

Lu = 0 in R

× S

. Then

u(r, t) =

Re(σ

±
j

)=0

(r, t) .

In particular, if

L has no spectral values on iR, then u ≡ 0.

Another consequence of the Laurent series representation is to patch together

solutions from both sides of the characteristic circle S

. More precisely we have the

following theorem.

Theorem

6.10. Suppose that

L has no spectral values in iR

∗

. Then we have

the following.

(1) If 0 is not a spectral value, then any bounded solution of

Lu = 0 in the

cylinder (

−R, R) × S

is continuous on the circle S

(2) If 0 is a spectral value (say with multiplicity 2), let g

(t) be the basic

solutions of

∗

with exponent 0. Then a bounded solution u of

Lu = 0 in

((

−R, 0) ∪ (0, R)) × S

is continuous on (

−R, R) × S

if and only if

2π

(θ)u(δ, θ)dθ = Re

2π

(θ)u(

−δ, θ)dθ

for some δ

∈ (0, R).

CHAPTER 7

The nonhomogeneous equation

Lu = F

After we extend the Cauchy integral formula to include the nonhomogeneous

case, we deﬁne an integral operator for the nonhomogeneous equation

Lu = F .

Throughout the remainder of this paper U will denote an open and bounded set
in

× S

whose boundary consists of ﬁnitely many simple, closed and piecewise

smooth curves.

7.1. Generalized Cauchy Integral Formula

The following generalization of the Cauchy integral formula will be used later.

Theorem

7.1. Suppose that F (r, t) is a function in U such that

∈ L

(U )

with p

≥ 1. If equation Lu = F has a solution u ∈ C

(U ), then

(7.1)

u(r, t)

−1

2π

∂

(r, t, ζ)u(ζ)

dζ

+ Ω

(r, t, ζ) u(ζ)

dζ

−

2π

(r, t, ζ)F (ζ) + Ω

(r, t, ζ) F (ζ)

dρdθ

Proof.

For δ > 0, let U

= U

\A(0, δ). Let z

∈ U and choose δ > 0 so that

∈ U

. Green’s identity (1.8) and arguments similar to those used in the proof of

the Cauchy integral formula show that

−2πu(z

) =

∂U

(r, t, ζ)u(ζ)

dζ

+ Ω

(r, t, ζ) u(ζ)

dζ

(r, t, ζ)F (ζ) + Ω

(r, t, ζ) F (ζ)

dρdθ

Since (F/r)

∈ L

with p

≥ 1, then the limits of the above integrals as δ → 0 give

(7.1).

For the adjoint operator, we have the following.

Theorem

7.2. Let v(ρ, θ)

∈ C

(U ) be such that

∗

∈ L

(U ) with p

≥ 1.

Then

(7.2)

v(ρ, θ)

−1

2π

∂

(z, ρ, θ)v(z)

+ Ω

(z, ρ, θ) v(z)

2π

(z, ρ, θ)

∗

v(z) + Ω

(z, ρ, θ)

∗

v(z)

drdt

7. THE NONHOMOGENEOUS EQUATION

Lu = F

Proof.

Notice that the kernels Ω

(z, ζ) and Ω

(z, ζ) satisfy

LΩ

−iλνΩ

+ c(t)Ω

and

LΩ

−iλνΩ

+ c(t)Ω

where L = λ

∂

∂t

− ir

∂

∂r

. Arguments similar to those used in the proofs of Theorems

6.4 and 7.1 lead to (7.2). The functions P

and P

used in the proof of Theorem 6.4

need now to be replaced by the functions Q

= Ω

+ Ω

and Q

−iΩ

+ i Ω

7.2. The integral operator T

We deﬁne the operator T and the appropriate L

-spaces in which it acts to

produce H¨

older continuous solutions. For an open set U

⊂ R

× S

as before and

such that S

⊂ ∂U, we denote by L

(U ) the Banach space of functions F (r, t) such

that

F (r, t)

is integrable in U with the norm

||F ||

p,a

F (r, t)

−1

drdt

1
p

Note that if Φ :

× S

−→ C

∗

is the diﬀeomorphism induced by the ﬁrst

integral z. That is, Φ(r, t) = r

, then F

∈ L

(U ) means that the push forward

F = F

◦ Φ

−1

satisﬁes

F (z)

∈ L

(Φ(U )).

We deﬁne the integral operator T by

(7.3)

T F (r, t) =

−1

2π

(r, t, ζ)F (ζ) + Ω

(r, t, ζ) F (ζ)

dρdθ

When

L has no purely imaginary spectral values, i.e. Spec(L) ∩ iR

∗

∅, we have

the following theorem.

Theorem

7.3. Assume Spec(

L) ∩ iR

∗

∅. Let U ⊂ A(0, R) be an open set as

above. The function T F deﬁned by (7.3) satisﬁes the followings.

1. There exist positive constants C and δ, independent on U and R, such

that for every (r, t)

∈ U

(7.4)

|T F (r, t)| ≤ CR

||F ||

p,a

for every F

∈ L

(U ) with p > 2/(1

− ν);

2. the function T F satisﬁes the equation

LT F = F ; and

3. the function T F is H¨

older continuous on U .

Furthermore, if 0 is not a spectral value, then T F (0, t)

≡ 0.

Proof.

We use the estimates on Ω

and Ω

of Theorem 5.1 to write

−2πu(r, t) = I

+ I

7.2. THE INTEGRAL OPERATOR T

where

= i

λν

− z

F (ζ)

dρdθ

−

λν

K(t, θ)L(z, ζ)F (ζ)

dρdθ

λν

c(t)L(z, ζ) + c(θ)

λν

L(z, ζ)

F (ζ)

dρdθ

(z, ζ)F (ζ) + C

(z, ζ) F (ζ)

dρdθ

We use the substitution ζ = Φ(ρ, θ) = ρ

iθ

to estimate I

. We ﬁnd

| ≤ r

aν

Φ(U )

F (ζ)

|ζ − z||ζ|

1+ν

dξdη ,

where we have set

F = F

◦ Φ

−1

and ζ = ξ + iη. Since

∈ L

(Φ(U )) with

p > 2/(1

− ν), and since Φ(U) is contained in the disc D(0, R

)

⊂ C, then H¨older

inequality can be used to show that there are constants C and δ so that

| ≤ r

aν

Furthermore these constants are independent on Φ(U ) and R.

Because of the

logarithmic type growth of L(z, ζ) and the boundedness of the functions C

and

, analogous arguments can be used to show that

| ≤ CR

||F ||

p,a

for k = 2, 3, 4.

Now we verify that u = T F solves

Lu = F in the sense of distributions. Let

∈ C

(U ) be a test function. The generalized Cauchy integral formula (7.2)

applied to ψ gives

(7.5)

ψ(ρ, θ) =

2π

(z, ρ, θ)

∗

ψ(z) + Ω

(z, ρ, θ)

∗

ψ(z)

drdt

The deﬁnition (7.3) of the operator T and estimate (7.4) give

2 < T F,

∗

ψ >

T F (z)

∗

ψ(z) + T F (z)

∗

ψ(z)

drdt

F (ζ)ψ(ζ) + F (ζ) ψ(ζ)

dρdθ

= 2 < F, ψ > .

This shows that

LT F = F .

Next, we prove that T F is H¨

older continuous. Since the equation is elliptic

away from the circle S

, it is enough to prove the regularity of T F on S

. For this,

we consider the case when 0 is not a spectral value. Then i

R ∩ Spec(L) = ∅ and

(0, t, ζ) = Ω

(0, t, ζ) = 0. Hence T F (0, t) = 0. Since T F satisﬁes

LT F = F , then

its pushforward V (z) = T F

◦ Φ

−1

(z) via the ﬁrst integral satisﬁes the generalized

CR equation

iλν

2iaz

−

c(z)

2iaz

−

F (z)

2iaz

where

c and

F are the pushforwards of c and F . We will use the classical results

on the CR equation (see [18]) to show that V is H¨

older continuous. We rewrite the

7. THE NONHOMOGENEOUS EQUATION

Lu = F

above equation as

(7.6)

G(z)

where

G(z) =

iλνz

2iaz

V (z)

−

c(z)

2iaz

V (z)

−

F (z)

2iaz

Note that since

c and V are bounded functions and since (

F /z)

∈ L

, then G

∈

(Φ(U )) with p > 2. The solution of (7.6) can then be written as V (z) =

W (z)

where W is the solution of the equation W

= G. We know that W is H¨

older

continuous and has the form

W (z) = H(z)

−

Φ(U )

G(ζ)

− z

dξdη

where H is a holomorphic function in Φ(U ).

Since V (z) = W (z)/z satisﬁes

V (0) = 0, then necessarily W (0) = 0 and it vanishes to an order > 1 at 0. Thus
|V (z)| ≤ K|z|

for some positive constants K and τ . This means that T F is H¨

older

continuous on S

Finally we consider the case when 0 is a spectral value of

L (say, with multi-

plicity 2). Let f

(t) and g

−j

(θ) be the basic solutions of

L and L

∗

with exponents

0. We have then

(0, t, ζ) =

(t)g

−j

(θ)

and

(0, t, ζ) =

(t) g

−j

(θ).

The value of T F on S

is found to be

T F (0, t) = A

(t) + A

−

(t) ,

where

−1

2π

−j

(θ)F (ζ)

dρdθ

Hence T F (0, t) solves the homogeneous equation

Lu = 0. Let

v(r, t) = T F (r, t)

− T F (0, t) .

The function v satisﬁes

Lv = F and v(0, t) = 0. The push forward arguments, used

in the case when 0 is not a spectral value, can be used again for the function v to
establish that

|v(r, t)| ≤ Cr

with τ and C positive.

In general, when

L has spectral values on iR, we can deﬁne 'Ω

and '

as in

(5.2) and (5.3) except that the terms corresponding to σ

that are in i

R are missing

from the sums. That is, if w

· · · , w

denotes the collection of basic solutions of

L with exponents in iR, and w

∗±

· · · , w

∗±

the corresponding collection of basic

solutions of the adjoint

∗

, then

(7.7)

'Ω

(z, ζ) = Ω

(z, ζ)

−

k=1

(z)w

∗±

(ζ) .

We deﬁne the modiﬁed operator '

T by

(7.8)

T F (r, t) =

−1

2π

'Ω

(r, t, ζ)F (ζ) + '

(r, t, ζ) F (ζ)

dρdθ

7.3. COMPACTNESS OF THE OPERATOR T

Arguments similar to those used in the proof of Theorem 7.3 establish the following
result.

Theorem

7.4. Let U

⊂ A(0, R) be as above. Then the function '

T F deﬁned by

(7.8) satisﬁes properties 1, 2, and 3 of Theorem 7.3 and '

T F (0, t) = 0.

7.3. Compactness of the operator T

Theorem

7.5. Suppose that

L has no spectral values in iR

∗

. Then, for p >

2/(1

− ν), the operator

T : L

p
a

(U )

−→ C

(U )

is compact.

Proof.

Let R > 0 be such that U

⊂ A(0, R). A function in L

(U ) can be

considered in L

(A(0, R)) by extending as 0 on A(0, R)

\U. Denote by T

the

operator T on the cylinder A(0, R) and set

F (r, t) = T

F (r, t)

− T

F (0, t).

Thus

F (r, t) =

−1

2π

A(0,R)

'Ω

(z, ζ)F (ζ) + '

(z, ζ) F (ζ)

dρdθ

where '

and '

are deﬁned by (5.2) and (5.3), respectively, except that the terms

corresponding to the spectral value σ

= 0 are missing. In particular, if 0 is not a

spectral value, then '

= Ω

and '

= Ω

. Note that '

(0, t) = 0 and '

(0, t) = 0.

The operator '

F satisﬁes the properties of Theorem 7.3 and '

F (0, t) = 0. To

show that T is compact, it is enough to show that '

is compact.

Let B

⊂ L

(U ) be a bounded set. We need to show that '

(B) is relatively

compact in C

(U ). Let M > 0 be such that

||F ||

p,a

≤ M for every F ∈ B. It

follows from Theorem 7.3 that '

(B) is bounded (by CR

M ). Now we show the

equicontinuity of '

(B). First along S

. For > 0, let r

> 0 be such that

M < (/2). We have then

F (z) = '

F (z) + '

A(r

,R)

F (z)

where '

A(r

,R)

denotes the integral operator over the cylinder A(r

, R). Let r

small enough so that

E =

max

r<(r

/2), r

<ρ<R

(

|'Ω

(r, t, ρ, θ)

| + |'Ω

(r, t, ρ, θ)

|) <

2M (πR

)

1/q

where q is such that

= 1. For r < r

/2, we have then

| '

F (r, t)

| ≤ Cr

≤

and

| '

A(r

,R)

F (r, t)

| ≤

A(r

,R)

(

|'Ω

| + |'Ω

|F (ζ)|

dρdθ

≤ E||F ||

p,a

≤

This estimate is obtained from H¨

older’s inequality and the above estimate on E.

Thus,

| '

F (r, t)

| ≤ and so '

B is equicontinuous on S

7. THE NONHOMOGENEOUS EQUATION

Lu = F

Next, let (r

, t

)

∈ U with r

< r

/4. Set z

= r

and z = r

. If

|z − z

| < r

/4, then r < r

/2 and the above argument gives

| '

F (z)

− '

F (z

)

| ≤ | '

(z)

| + | '

)

| ≤ 2,

∀F ∈ B.

Finally, suppose that r

> r

/4. Let b be such that 0 < b < r

/4. We write

F (z)

− '

F (z

) = '

F (z)

− '

F (z

) + '

A(b,R)

F (z)

− '

A(b,R)

F (z

After using H¨

older’s inequality we obtain

| '

F (z)

− '

F (z

)

| ≤ CS(b)||F ||

p,a

where

S(b) = max

(

|'Ω

(z, ζ)

− 'Ω

, ζ)

| + |'Ω

(z, ζ)

− 'Ω

, ζ)

|) ,

and where the maximum is taken over the set P of points satisfying ρ < b,

|r−r

| <

b, r > r

/4 and r

> r

/4. The continuity of the kernels in the region ρ < b and

r > r

/4 implies that if b is small enough, then S(b) < /(2M C) and consequently

| '

F (z)

− '

F (z

)

| ≤

Finally, for '

A(b,R)

F , it suﬃces to notice that it solves the equation

Lu = F in the

cylinder A(r

/4, R). In this cylinder, the equation is elliptic and the classical theory

of generalized analytic function ([18] Chapter 7) implies that the family '

A(b,R)

is equicontinuous.

CHAPTER 8

The semilinear equation

In this chapter, we make use of the operator T and of its modiﬁed version,

through the kernels Ω

j,1

and Ω

j,2

(deﬁned in Chapter 5), to establish a correspon-

dence between the solutions of the homogeneous equation

Lu = 0 and the solutions

of a semilinear equation.

Theorem

8.1. Assume that

L has no spectral values in iR

∗

. Let G(u, r, t) be a

bounded function deﬁned in

C × A(0, R

), for some R

> 0, and let τ > aν. Then,

there are R > 0 and a one to one map between the space of continuous solutions
of the equation

Lu = 0 in A(0, R) and the space of continuous solutions of the

equation

(8.1)

Lu = r

|u|G(u, r, t) .

Furthermore, if v is a bounded solution of (8.1) in a cylinder A(0, R), then v is
continuous up to the circle S

Proof.

First note that since τ > aν, then r

∈ L

(A(0, R)), with p satisfying

p > 2/(1

−ν), and ||r

p,a

= C

with C

and δ

positive. Consider the operator

P : C

(A(0, R))

−→ C

(A(0, R)) ;

P(f) = T

|f|G(f, r, t))

where, as before, T

denotes the integral operator on the cylinder A(0, R). Note

that since G is a bounded function, then

P is well deﬁned. It follows from the

properties of T given in Theorem 7.3, from the boundedness of G, and from r

∈ L

that the operator

P satisﬁes

L(P(f)) = r

|f|G(f, r, t)

and

|P(f)(r, t)| ≤ CR

||r

|f|G(f, r, t)||

p,a

≤ C

||f||

∀f ∈ C

(A(0, R))

with C

and δ

positive. Hence, if R > 0 is small enough,

||P|| ≤ C

< 1, and

is thus a contraction. Let

F = (I − P)

−1

. The operator

F realizes the one to one

mapping between the space of continuous solutions of

Lu = 0 and those of equation

(8.1).

Now, we show that if v a bounded solution of (8.1) in a cylinder A(0, R), then

it is continuous. For a bounded solution v, the function r

|v|G(v, r, t) is bounded

and is in L

(A(0, R)). Consequently,

P(v) is continuous up to the boundary S

The function u = v

− P(v) is a bounded solution of Lu = 0 and so it is contin-

uous up to S

(Remark 6.6). It follows that v = u +

P(v) is also continuous up

to S

8. THE SEMILINEAR EQUATION

Let σ

= σ

(or σ

= σ

−

) be a spectral value of

L such that Re(σ

) > 0.

Consider the Banach spaces r

(A(0, R)) and r

(A(0, R)) deﬁned as follows:

∈ r

p
a

(A(0, R))

∈ L

p
a

and

∈ r

(A(0, R))

∈ C

(A(0, R))

∩ L

∞

(A(0, R)) .

The norms in these spaces are deﬁned by

||f||

p,a,σ

p,a

and

||g||

0,σ

Consider the operator T

deﬁned by

(8.2)

F (r, t) =

−1

2π

A(0,R)

j,1

(z, ζ)F (ζ) + Ω

j,2

(z, ζ) F (ζ)

dρdθ

where Ω

j,1

and Ω

j,2

denote the modiﬁed kernels deﬁned in (5.11) and (5.12). Note

that the estimates of Theorem 5.4 on the modiﬁed kernels imply that T

F is in

(A(0, R)) when F is in r

(A(0, R)). Arguments similar to those used in

the proof of Theorem 7.3 can be used to establish the following theorem.

Theorem

8.2. For p > 2, the operator

: r

p
a

(A(0, R))

−→ r

(A(0, R))

satisﬁes

(8.3)

F (r, t) = F (r, t)

and

||T

0,σ

≤ CR

||F ||

p,a,σ

where C and δ are positive constants.

Two functions u and v deﬁned in the cylinder A(0, R) are said to be similar if

u/v is continuous in A(0, R) and there exist positive constants C

and C

such that

≤

u(r, t)

v(r, t)

≤ C

∀(r, t) ∈ A(0, R) .

Theorem

8.3. Let

L, τ, and G be as in Theorem 8.1. Then there exists R > 0

such that each continuous solution of

Lu = 0 in A(0, R) is similar to a continuous

solution of equation (8.1).

Proof.

Let u be a continuous solution of

Lu = 0 on A(0, R). Let μ ≥ 0 be

the order of u along S

. If μ > 0, then μ = Re(σ

) for some spectral value σ

Assume that σ

−

= σ

. Then it follows from the Laurent series expansion

that u is similar to a linear combination

(r, t) = a

−

(r, t) + a

+
j

(r, t)

of the basic solutions w

+
j

and w

−

in a cylinder A(0, R

) with 0 < R

< R. Consider

the operator

: r

(A(0, R))

−→ r

(A(0, R))

deﬁned by

(f ) = T

|f|G(f, r, t)).

It follows from the hypotheses on τ , on G, and from Theorem 8.2 that

(f ) = r

|f|G(f, r, t) and ||P

(f )

0,σ

≤ KR

||f||

0,σ

8. THE SEMILINEAR EQUATION

for some positive constants K and δ. In particular,

is a contraction, if R is small

enough. Hence, the function v = (I

− P

)

−1

(u) is a solution of equation (8.1) and

it is also similar to u

If μ = 0 (then we are necessarily in the case where 0 is a spectral value), let

be the resolvent of

P used in the proof of Theorem 8.1. Then v = F(u) is similar

to u and solves (8.1).

A direct consequence of Theorem 8.1 and Theorem 6.7 is the following unique-

ness result for the solutions of (8.1).

Theorem

8.4. Suppose that the spectral values of

L satisfy the following con-

dition

Re(σ

) = Re(σ

) =

⇒ σ

= σ

∀σ

, σ

∈ Spec(L) .

Let u be a bounded solution of (8.1) in a cylinder A(0, R). Suppose that there is a
sequence of points (r

, t

)

∈ A(0, R) such that r

→ 0 and u(r

, t

) = 0 for every

∈ Z

. Then u

≡ 0.

CHAPTER 9

The second order equation: Reduction

This chapter deals with the second order operator P = LL + Re(aL). We show

that the equation P u = F , with u and F real-valued, can be reduced to an equation
of the form

Lu = G.

As before, let λ = a + ib

∈ R

+ i

R, L be the vector ﬁeld given by (1.1) and

let β(t)

∈ C

C), with m ≥ 2, satisﬁes

(9.1)

2πi

2π

β(t)dt = k

∈ Z .

Consider the second order operator P deﬁned as

(9.2)

P = LL + λβ(t)L + λβ(t) L .

Then,

P u =

|λ|

− 2bru

+ r

|λ|

(β + β)u

+ [1

− i(λ β − λβ)]ru

Note that P is elliptic except along the circle S

{0}×S

, and that P u is

R-valued

when u is

R-valued.

With the operator P we associate a ﬁrst order operator

L and show that the

equation P u = F , with F real-valued, is equivalent to an equation of the form
Lw = G. Let

(9.3)

B(t) = exp

β(s)ds .

It follows from (9.1) that B is periodic with Ind(B) =

−k and satisﬁes LB = λβ B.

Deﬁne the function c(t) by

(9.4)

c(t) =

−λ β(t)

B(t)

−λ β(t) exp

(β(s)

− β(s))ds

Note that the function B satisﬁes also the equation

(9.5)

LB =

−c(t)

(t)

B(t)

To each

R-valued function u, we associate the C-valued function w deﬁned by

w(r, t) = B(t)Lu(r, t) .

We will refer to w as the L-gradient of u with respect to P . To the operator P we
associate the ﬁrst order operator

L deﬁned by

(9.6)

Lw = Lw − c(t)w ,

where c(t) is given by (9.4). We have the following proposition.

9. THE SECOND ORDER EQUATION: REDUCTION

Proposition

9.1. Suppose that F is an

R-valued function and u(r, t) is R-

valued and solves the equation

(9.7)

P u(r, t) = F (r, t)

in the cylinder A(0, R). Then its L-gradient w satisﬁes the equation

(9.8)

Lw(r, t) = B(t)F (r, t).

Conversely, if w is a solution of (9.8) in A(0, R), then there is an

R-valued function

u deﬁned in A(0, R) that solves (9.7) and whose L-gradient is w. More precisely,
the function u can be deﬁned by

(9.9)

u(r, t) = Re

(r,t)

)

w(ρ, θ)

B(θ)

dζ

iaζ

where ζ = ρ

iθ

and the integration is taken over any simple curve in A(0, R) that

joins the ﬁxed point (r

, t

) to the point (r, t).

Proof.

Suppose that u is

R-valued and solves (9.7). By using (9.4) and (9.5),

we see that its L-gradient satisﬁes

Lw = L(BLu)

= BLLu + LBLu = BF

− λβBLu

= BF

− λ β

(BLu) = BF + cw.

Thus w solves (9.8). Conversely, suppose that w solves (9.8). Let (r

, t

)

∈ A(0, R)

and consider the function u(r, t) deﬁned by (9.9).

We need to verify that the

integral is path independent. Let U be a relatively compact subset of A(0, R)
whose boundary consists of simple closed curves.

It follows from the proof of

Green’s identity (1.8) and (9.5) that

∂U

w(ζ)

B(θ)

dζ

iaζ

dζdζ

|ζ|

−

idρdθ

F +

w +

idρdθ

Since F is

R-valued, then the real part of the above integral is zero and the function

u is well deﬁned. That u satisﬁes (9.7) follows easily by computing the derivatives
u

and u

from (9.9) to obtain Lu = w/B and then using (9.8) to get (9.7).

CHAPTER 10

The homogeneous equation P u = 0

We use the reduction given in Proposition 9.1 to obtain properties of the solu-

tions of the equation P u = 0 from those of their L-gradients w. In particular, series
representation for u in a cylinder is derived. Under an assumption on the spectrum
of

L, we prove a maximum principle for the equation P u = 0: The extreme values

of u can occur only on the distinguished boundary ∂

U . It should be mentioned

that many results in this section and the next are close to those obtained, in [11].
There, the operator in

has its principal part of the particular form (x

+ y

)Δ,

where Δ is the Laplacian. Such an operator, when written in polar coordinates has
the form (9.2) with the vector ﬁeld L having the invariant λ = 1.

10.1. Some properties

The following simple properties for the solutions u will be needed . We start

by considering the possibility of the existence of radial solutions.

Proposition

10.1. The equation P u = 0 has radial solutions u = u(r) if and

only if the coeﬃcient β has the form

(10.1)

β(t) =

p(t)

− ik

where k

∈ Z and p(t) is R-valued and such that

2π

p(t)dt = 0 .

In this case, the radial solutions have the form

(10.2)

u(r) =

log r + C

if k = 0

2ak

+ C

if k

= 0

where C

, C

are arbitrary constants. The corresponding L-gradients are

w(r, t) = iC

B(t),

when

k = 0

and

w(r, t) = 2iakC

2ak

B(t),

when

= 0 ,

where

B(t) = e

ikt

exp

p(s)ds

Moreover, w(r, t) is a basic solution of

L with character (2ak, k).

Proof.

If u = u(r) solves P u = 0, then

(r) + (1

− i(λ β(t) − λ β(t)))ru

(r) = 0 .

Hence, i(λ β(t)

− λ β(t)) is a real constant. If we set β(t) = p(t) + iq(t) with

p and q real-valued, then aq(t)

− bp(t) = M, with M ∈ R constant. It follows

10. THE HOMOGENEOUS EQUATION P u = 0

from hypothesis (9.1) that M =

−ak with k ∈ Z and that the average of p is

zero. This gives aq(t) = bp(t)

− ak and consequently β has the form (10.1). For

such a coeﬃcient β, the radial solutions are easily obtained from the diﬀerential
equation.

Remark

10.2. Note that if u = u(t) (independent on r) solves the equation

P u = 0, then u is necessarily constant.

The following lemma will be used in the proof of the next proposition.

Lemma

10.3. Let u(r, t) be a solution of P u = 0 in the cylinder A(0, R) and

let w = BLu be its L-gradient. If

λw(r, t)

iaB(t)

≡ 0 ,

then u is constant.

Proof.

Let (r

, 0) be a ﬁxed point in the cylinder A(0, R). Let u be as in the

lemma and let w = BLu. The function

v(r, t) = Re

Γ(r,t)

w(ζ)

B(θ)

dζ

iaζ

where Γ(r, t) is any piecewise smooth curve that joins the point (r

, 0) to the point

(r, t), solves P v = 0 (Proposition 9.1). We choose Γ as Γ = Γ

∪ Γ

, where

{(r

, st), 0

≤ s ≤ 1} and Γ

{((1 − s)r

+ sr, t), 0

≤ s ≤ 1}.

With this choice of Γ and with the hypothesis of the lemma on the gradient w, the
integral over Γ

is 0 and the expression for v reduces to

v(r, t) =

w(r

, st)

B(st)

ds .

Hence, the function v depends only on the variable t and since it solves P v = 0, then
v is constant (Remark 10.2). Consequently, w = BLv = 0. This means Lu = 0.
Since u is

R-valued, then Lu = 0 and so u is constant.

Proposition

10.4. Suppose that u

∈ C

(A(0, R)) solves P u = 0, then its L-

gradient w satisﬁes, w

∈ C

(A(0, R)

∪ S

) and w(0, t)

≡ 0. Moreover, u is constant

along S

Proof.

Since P is elliptic for r

= 0, then we need only to verify the continuity

of w up to S

and its vanishing there. As a solution of

Lw = 0, the function w has

a Laurent series expansion (Theorem 6.2)

w(r, t) =

∈Z

(r, t)

where w

are the basic solutions of

L. Let τ ∈ R be the order of w along S

(that

the order τ is ﬁnite is a consequence of the continuity of u up to S

). We are going

to show that τ > 0. Let w

· · · , w

be the collection of all basic solutions with

10.2. MAIN RESULT ABOUT THE HOMOGENEOUS EQUATION P u = 0

order τ along S

. That is, w

is a basic solution with Char(w

) = (σ

, j

) and

such that the exponent satisﬁes Re(σ

) = τ . We have then

w(r, t) =

k=1

(r, t) + o(r

) = w

(r, t) + o(r

It follows from Lemma 10.3 that for each k, Re(λw

/iaB)

≡ 0. Let t

∈ R be such

that

λw

(r, t

)

iaB(t

)

= 0,

k = 1,

· · · , N.

Let r

< R be ﬁxed. By using integration over the segment from (r

, t

) to (r, t

we ﬁnd

u(r, t

)

− u(r

, t

) =

λw

((1

− s)r

+ sr, t

)

iaB(t

)

− r

)ds

− s)r

+ sr

+ o(r

Recall that each basic solution w

· · · , w

has an exponent σ

= τ + iβ

and so

− r

)λw

((1

− s)r

+ sr, t

)

((1

− s)r

+ sr)iaB(t

)

ds =

⎧

⎨
⎩

O(r

)

if τ

= 0

O(log r)

if τ = 0, β

= 0

O(r

iβ

)

if τ = 0, β

= 0.

From these estimates and the above integral, we deduce that in order for u(r, t

)

−

u(r

, t

) to have a limit as r

→ 0, it is necessary that τ > 0. For such τ, w(0, t) = 0,

and u(0, t) is constant.

As a consequence of the proof of Proposition 10.4, we have the following propo-

sition.

Proposition

10.5. Suppose that

L has no spectral values on iR

∗

. If u

∈

∞

(A(0, R)) solves P u = 0, then u is continuous up to the boundary S

and it is

constant on S

10.2. Main result about the homogeneous equation P u = 0

We use the basic solutions of the associated operator

L to construct 2π-periodic

functions q

(t) and establish a series expansion of the continuous solutions u.

Let

{σ

}

∈Z

be the spectrum of the associated operator

L and w

be the

corresponding basic solutions. Recall that if σ

∈ R, then w

= r

±
j

(t) with

Ind(f

) = j and if σ

∈ C\R, then σ

−

= σ

and

+
j

(r, t) = r

(t) + r

(t) ,

−

(r, t) = i

(t)

− r

(t)

with

|φ

| > |ψ

| and Ind(φ

) = j. Deﬁne the functions q

(t) as follows. For

∈ R

∗

(10.3)

(t) =

iaσ

(t)

B(t)

10. THE HOMOGENEOUS EQUATION P u = 0

and for σ

∈ C\R,

(10.4)

(t) =

iaσ

λφ

(t)

B(t)

−

λ ψ

(t)

B(t)

−

(t) =

aσ

λφ

(t)

B(t)

λ ψ

(t)

B(t)

It follows from Theorem 4.1 that the asymptotic behaviors of q

are

(t) =

ijt

iajB(t)

+ O(j

−2

)

and

−

(t) =

ijt

ajB(t)

+ O(j

−2

We have the following representation theorem.

Theorem

10.6. If u

∈ C

(A(0, R)) is a solution of P u = 0, then u is constant

on S

and it has the series expansion

(10.5)

u(r, t) = u

Re(σ

±
j

)>0

±
j

(t)

where the functions q

are deﬁned in (10.3) and (10.4), and where u

∈ R.

Proof.

It follows from Proposition 10.4 that u is constant on S

. Hence, by

using integration over the segment from (0, t) to (r, t), we obtain

u(r, t)

− u(0, 0) = u(r, t) − u(0, t) = Re

λw(sr, t)

iaB(t)

where w is the L-gradient of u. The function w, being a solution of

Lw = 0, has a

series expansion

w(r, t) =

Re(σ

±
j

)>0

(r, t).

For the function u we have then

u(r, t) = u(0, 0) +

Re(σ

±
j

)>0

λw

(sr, t)

iaB(t)

Now for σ

∈ R, we have w

(r, t) = r

±
j

(t) and

λw

(sr, t)

iaB(t)

= r

±
j

iaσ

(t)

B(t)

= r

±
j

(t) .

For σ

= σ

−

= σ

∈ C\R, we have

λw

+
j

(sr, t)

iaB(t)

(rs)

λφ

(t)

iaB(t)

+ (rs)

λψ

(t)

iaB(t)

λφ

(t)

iaB(t)

λψ

(t)

iaB(t)

From this and (10.4), we get

λw

+
j

(sr, t)

iaB(t)

= Re

(t)

10.3. A MAXIMUM PRINCIPLE

A similar relation holds for the integral of w

−

and the series expansion (10.5)

follows.

Remark

10.7. A consequence of this theorem and of the asymptotic expansion

of the spectral values σ

, given in Theorem 4.1, imply that the number λ is an

invariant for the operator P in the following sense: Suppose that

= L

+ λ

(t)L

+ λ

(t) L

is generated by a vector ﬁeld L

with invariant λ

= a

+ ib

∈ R

+ i

R and such

that for every k

∈ Z

, there is a diﬀeomorphism, Φ

, in a neighborhood of the

circle S

such that that Φ

∗

P is a multiple of P

, then λ = λ

10.3. A maximum principle

We use the series representation of Theorem 10.6 to obtain a maximum principle

when the spectrum satisﬁes a certain condition.

Recall that the function B(t) satisﬁes Ind(B) =

−k, where k ∈ Z is deﬁned by

(9.1). We will say that the operator P satisﬁes hypothesis

H if the spectrum of L

satisﬁes the following conditions:

: Re(σ

)

≤ 0 =⇒ j ≤ −k.

: Re(σ

) = Re(σ

) =

⇒ σ

= σ

Thus P satisﬁes

H means that the projection of Spec(L) into R is injective and that

the basic solutions w of

L with positive orders have winding numbers Ind(w) > k.

Theorem

10.8. Suppose that the operator P satisﬁes

H. Let U ⊂ R

× S

open, bounded, and such that A(0, R)

⊂ U for some R > 0. If u ∈ C

(U ) satisﬁes

P u = 0, then the value of u on S

is not an extreme value of u. Thus the maximum

and minimum of u occur on ∂U

Proof.

Let τ > 0 be the order along S

of the L-gradient of u. It follows from

Theorem 10.6 that

(10.6)

u(r, t)

− u(0, 0) =

Re(σ

±
j

)=τ

±
j

(t)

+ o(r

We consider two cases depending on whether τ is a spectral value of

L or τ is only

the real part of a spectral value. Note that it follows from

that the sum in (10.6)

consists of either one term or two terms. It has one term, if τ is a spectral value
with multiplicity one. It has two terms if τ is a spectral value with multiplicity two
or if τ is not a spectral value.

If τ is spectral value (say with multiplicity 2), then the corresponding basic

solutions have the form r

(t) with winding number j >

−k (by condition H

After replacing, in (10.6), the functions q

by their expressions given in (10.3), we

ﬁnd that

u(r, t)

− u(0, 0) = r

iaτ

+
j

(t) + c

−

(t)

B(t)

+ o(r

10. THE HOMOGENEOUS EQUATION P u = 0

Recall that the functions f

and f

−

are

R-independent solutions of the diﬀerential

equation (2.3). Thus, c

+
j

+ c

−

has winding number j and consequently

Ind

iaτ

+
j

(t) + c

−

(t)

B(t)

= j + k > 0

(we have used the fact that Ind(B) =

−k). Since the winding number is positive,

the real part changes sign. This implies that u(r, t)

− u(0, 0) changes sign (for r

small) and u(0, 0) is not an extreme value. The proof for the case when τ is a
spectral value with multiplicity one is similar.

If τ is not a spectral value, then there is a unique spectral value σ = τ + iμ

with μ

= 0. The corresponding basic solutions w

have winding number j >

−k.

After substituting, in (10.6), the functions q

by their expressions given in (10.4)

we ﬁnd

u(r, t)

− u(0, 0) = r

iμ

iaσ

Dλ

(t)

B(t)

− Dλ

(t)

B(t)

+ o(r

) ,

where D = c

+ic

−

and where φ

and ψ

are the components of the basic solutions.

We have

|φ

| > |ψ

| and Ind(φ

) = j. The same argument as before shows that

u(r, t)

− u(0, 0) changes sign, as the real part of a function with winding number

j + k > 0.

Remark

10.9. If the condition

H is not satisﬁed, then equation P u = 0 might

have solutions with extreme values on S

. Consider for example the case in which

the function β(t) is given by (10.1) with k = 1 (In this example we have Ind(B) = 1).
The operator P does not satisfy

. Indeed, 2a is a spectral value, corresponding

to the basic solution r

B(t), with winding number 1. The corresponding basic

solution r

has minimum value 0 and it is attained on S

CHAPTER 11

The nonhomogeneous equation P u = F

We construct here integral operators for the equation P u = F . A similarity

principle between the solutions of P u = 0 and those of a semilinear equation is
then obtained through these operators.

Let '

and '

be the functions given by (7.7). Deﬁne the function S(z, ζ) by

(11.1)

S(z, ζ) = Re

−λ

2πaiB(t)

'Ω

(sr, t, ζ)B(θ) + '

(sr, t, ζ)B(θ)

and the integral operator

K by

(11.2)

KF (r, t) =

A(0,R)

S(r, t, ρ, θ)F (ρ, θ)

dρdθ

We have the following theorem.

Theorem

11.1. If p > 2 and R > 0, then there exist positive constants C and

δ such that

K : L

(A(0, R))

−→ C

(A(0, R)) has the following properties

P (

KF ) = F, KF (0, t) = 0, and |KF (r, t)| ≤ CR

||F ||

p,a

Proof.

For an

R-valued function F ∈ L

(A(0, R)), with p > 2, consider

(11.3)

(B(t)F (z)) =

−1

2π

A(0,R)

'Ω

(z, ζ)B(θ) + '

(z, ζ)B(θ)

F (ζ)

dρdθ

We know, from Theorem 7.4, that '

(BF )

∈ C

(A(0, R)) satisﬁes

L '

(BF ) = BF,

(BF )(0, t) = 0,

and

| '

(BF )(r, t)

| ≤ CR

||BF ||

p,a

≤ C||B||

||F ||

p,a

∀F ∈ L

(A(0, R))

for some positive constants C

and δ. Furthermore, it follows from (11.1), (11.2),

and (11.3) that

(11.4)

K(F )(r, t) = Re

(BF )(sr, t)

B(t)

Then, from Proposition 9.1, we conclude that '

(BF ) is the L-gradient of

K(F ).

The conclusion of the theorem follows from (11.4) and from the properties
of '

To established a similarity principle between the solutions of P u = 0 and those

of an associated semilinear equation, we need to use the modiﬁed kernels of Chapter

11. THE NONHOMOGENEOUS EQUATION P u = F

5. For j

∈ Z, let Ω

j,1

and Ω

j,2

be the kernels given by (5.11) and (5.12). Deﬁne S

(11.5)

(z, ζ) = Re

−λ

2πaiB(t)

j,1

(sr, t, ζ)B(θ) + Ω

j,2

(sr, t, ζ)B(θ)

and the operator

(11.6)

F (r, t) =

A(0,R)

(r, t, ρ, θ)F (ρ, θ)

dρdθ

The operators T

, deﬁned in (8.2), and

are related by

(11.7)

F (r, t) = Re

(BF )(sr, t)

B(t)

The operator

acts on the Banach space r

±
j

(A(0, R)), deﬁned in Chapter 8,

and produces continuous functions that vanish along S

. More precisely, deﬁne the

Banach space r

±
j

E(A(0, R)) to be the set of functions v(r, t) that are in C

(A(0, R))

such that

±
j

and

±
j

are bounded functions in A(0, R). The norm of v is

σ±

(

( v

±
j

(

( Lv

±
j

(

The next theorem can be proved by using Theorem 8.2 and arguments similar to
those used in the proof of Theorem 11.1.

Theorem

11.2. The operator

: r

±
j

p
a

(A(0, R))

−→ r

±
j

E(A(0, R))

satisﬁes P

F = F and

σ±

≤ CR

p,a,σ

±
j

for some positive constants C and δ.

Let f

(r, t), f

(r, t), and f

(r, t) be bounded functions in A(0, R) and let

(r, t, u, w) and g

(r, t, u, w) be bounded functions in A(0, R)

× R × C. Deﬁne

the function H by

(11.8)

H(r, t, u, w) = uf

+ wf

|u|

1+α

|w|

1+α

with α > 0. For > 0, consider the semilinear equation

(11.9)

P u = r

Re (H(r, t, u, Lu)) .

We have the following similarity result between the solutions of (11.9) and those of
the equation P u = 0.

Theorem

11.3. For a given function H deﬁned by (11.8), there exists R > 0

such that, for every u

∈ C

(A(0, R)) satisfying P u = 0 and u = 0 on S

, there

exists a function m

∈ C

(A(0, R)) satisfying

≤ m(r, t) ≤ C

∀(r, t) ∈ A(0, R)

with C

and C

positive constants, such that the function v = mu solves equation

(11.9).

11. THE NONHOMOGENEOUS EQUATION P u = F

Proof.

Let u be a solution of P u = 0 with order τ > 0 along S

. Then there is

∈ Spec(L) such that τ = Re(σ

). Hence, u

∈ r

±
j

E(A(0, R

)) for some R

> 0.

Consider the operator

Q : r

±
j

E(A(0, R

))

−→ r

±
j

E(A(0, R

))

given by

Qv = K

Re(H(r, t, v, Lv))). It follows from (11.8) that the function

Re(H(r, t, v, Lv)) is in the space r

±
j

. Now, Theorems 11.2, 8.2, and relation

(11.7), imply that

Qv(r, t) = r

Re(H(r, t, v, Lv)),

Qv(r, t) = T

[B(t)r

Re(H(r, t, v, Lv))] .

Consequently,

σ±

≤ CR

σ±

. If R

is small enough, we have

Q < 1,

and we can deﬁne the resolvent

F = (I − Q)

−1

. It is easily checked that for the

solution u of P u = 0 as above, the function v =

F(u) solves equation (11.9) and

m = u/v is bounded away from 0 and

∞.

CHAPTER 12

Normalization of a Class of Second Order

Equations with a Singularity

This section deals with the normalization of a class of second order operators

whose coeﬃcients vanish at a point. To such an operator, a complex number

λ = a + ib

∈ R

+ i

R is invariantly associated. It is then shown that the operator D

is conjugate, in a punctured neighborhood of the singularity, to a unique operator
P given by (9.2). The properties of the solutions of the equations corresponding to

D are, thus, inherited from the solutions of the equations for P studied in Chapters
10 and 11.

Let

D be the second order operator given in a neighborhood of 0 ∈ R

(12.1)

Du = a

+ 2a

+ a

where the coeﬃcients a

· · · , a

are C

∞

, real-valued functions vanishing at 0,

with a

nonnegative, and

(12.2)

≤

(x, y)a

(x, y)

− a

(x, y)

+ y

)

≤ C

for some positive constants C

< C

. It follows in particular that a

and a

vanish

to second order at 0. Let A and B be the functions deﬁned for (x, y)

= 0 by

(12.3)

A(x, y)

+ y

)

− a

− 2a

xy + a

B(x, y)

− a

)xy + a

− y

)

− 2a

xy + a

Note that it follows from (12.2) that these functions are bounded and A is positive.
Let

(12.4)

μ =

2π

lim

→0

A(x, y)

− iB(x, y)

+ y

(xdy

− ydx) ,

where C

denotes the circle with radius ρ and center 0. We will prove that μ

∈

+ i

R is well deﬁned and it is an invariant for the operator D.

We will be using the following normalization theorem for a class of vector ﬁelds

in a neighborhood of a characteristic curve.

Theorem

12.1. Let X be a C

∞

complex vector ﬁeld in

satisfying the fol-

lowing conditions in a neighborhood of a smooth, simple, closed curve Σ:

(ı) X

∧ X

= 0 for every p /∈ Σ;

(ıı) X

∧ X

vanishes to ﬁrst order for p

∈ Σ; and

(ııı) X is tangent to Σ.

NORMALIZATION OF A CLASS OF SECOND ORDER EQUATIONS

Then there exist an open tubular neighborhood U of Σ, a positive number R, a
unique complex number λ

∈ R

+ i

R, and a diﬀeomorphism

Φ : U

−→ (−R, R) × S

such that

∗

X = m(r, t)

∂

∂t

− ir

∂

∂r

where m(r, t) is a nonvanishing function. Moreover, when λ

∈ Q, then for any

given k

∈ Z

, the diﬀeomorphism Φ and the function m can be taken to be of class

This normalization Theorem was proved in [8] when λ

∈ C\R. When λ ∈ R,

only a C

-diﬀeomorphism Φ is achieved in [8]. A generalization is obtained by

Cordaro and Gong in [4] to include C

-smoothness of Φ when λ

∈ R\Q. It is also

proved in [4], that, in general, a C

∞

-normalization cannot be achieved.

We will be using polar coordinates x = ρ cos θ, y = ρ sin θ and we will denote

this change of coordinates by Ψ. Thus,

Ψ :

\0 −→ R

× S

Ψ(x, y) = (ρ, θ) .

Theorem

12.2. Let

D be the second order operator given by (12.1) whose co-

eﬃcients vanish at 0 and satisfy condition (12.2). Then there is a neighborhood U
of the circle

{0} × S

in [0,

∞) × S

, a positive number R, a diﬀeomorphism

Φ : U

−→ [0, R) × S

sending

{0} × S

onto itself, such that

(12.5)

(Φ

◦ Ψ)

∗

D = m(r, t)

LL + Re(β(r, t)L)

where m, β are diﬀerentiable functions with m(r, t)

= 0 for every (r, t) and

L = λ

∂

∂t

− ir

∂

∂r

with λ =

and μ given by (12.4). Moreover, if the invariant μ /

∈ Q, then for every

∈ Z

, the diﬀeomorphism Φ, and the functions m, and β can be chosen to be of

class C

Proof.

We start by rewriting

D in polar coordinates:

(12.6)

Du = P u

θθ

+ 2N u

ρθ

+ M u

ρρ

+ Qu

+ T u

where

sin

− 2a

sin θ cos θ + a

cos

−a

sin θ cos θ + a

(cos

− sin

θ) + a

cos θ sin θ

= a

cos

θ + 2a

sin θ cos θ + a

sin

− 2a

sin θ cos θ + a

cos

+ a

cos θ + a

sin θ

sin θ cos θ+a

(sin

−cos

θ)

−a

sin θ cos θ

−

sin θ+a

cos θ).

Condition (12.2) implies that there is a constant C

> such that

M (ρ, θ)

≥ C

and

P (ρ, θ)

≥ C

∀(ρ, θ).

NORMALIZATION OF A CLASS OF SECOND ORDER EQUATIONS

We deﬁne the following C

∞

functions (of (ρ, θ))

ρP

In terms of these function, (12.2) takes the form

(12.7)

(ρ, θ)

− N

(ρ, θ)

≥ C

∀(ρ, θ) ∈ [0, R

]

× S

and (12.6) becomes

(12.8)

= u

θθ

+ 2ρN

ρθ

+ ρ

ρρ

+ ρQ

+ T

Let X be the C

∞

complex vector ﬁeld deﬁned by

(12.9)

X =

∂

∂θ

− ρg(ρ, θ)

∂

∂ρ

with g = N

+ i

− N

. Although we will use X for ρ

≥ 0, the vector ﬁeld X

is deﬁned in a neighborhood of

{0} × S

R × S

. By using X and its complex

conjugate X, we ﬁnd that

(12.10)

XXu

= u

θθ

+ 2ρN

θρ

+ ρ

ρρ

+ ρf u

where

f =

X(ρg)

−|g|

+ X(g).

We also have

(12.11)

ρu

− Xu

− g

and

gXu

− gXu

− g

It follows from (12.8), (12.10) and (12.11) that

(12.12)

= XXu

−

− Q − 1 + gT

− g

Xu +

− Q − 1 + gT

− g

Xu.

Since the coeﬃcients of

D and the function u are R-valued, then the right hand side

of (12.12) is real valued and can be written as

(12.13)

= XXu + X Xu + B(ρ, θ)Xu + B(ρ, θ) Xu

with

B(ρ, θ) =

−

f + f

− 2Q

+ 2gT

− g

Now, for the vector ﬁeld X, we have

∧ X = ρ(g − g)

∂

∂θ

∧

∂

∂ρ

−2iρ

− N

∂

∂θ

∧

∂

∂ρ

and so X satisﬁes the conditions of Theorem 12.1 and therefore it can be normalized.
In our setting, the invariant λ is given by (see [8]) λ = 1/

μ where

μ =

2π

(0, θ)

− N

(0, θ)

− iN

(0, θ)

dθ = μ,

and where μ is given by (12.4). Hence, there is a diﬀeomorphism Φ deﬁned in a
neighborhood of ρ = 0 in

R × S

onto a cylinder (

−R, R) × S

such that Φ

∗

X =

m(r, t)L with L as in the Theorem and m a nonvanishing function. Finally, it follows

NORMALIZATION OF A CLASS OF SECOND ORDER EQUATIONS

from this normalization of X that, in the (r, t) coordinates, expression (12.13)
becomes

(12.14)

= 2

|m|

LLu + (mB + m Lm) Lu + (m B + mLm) Lu .

This completes the proof of the theorem.

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ematics, 1984 Mathematics Road, University of British Columbia, Vancouver, BC, Canada V6T 1Z2;
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ment of Mathematics, CUNY Queens College and Graduate Center, 65-30 Kissena Blvd., Flushing, NY
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Selected Titles in This Series

1022 Maurice Duits, Arno B. J. Kuijlaars, and Man Yue Mo, The Hermitian two

matrix model with an even quartic potential, 2012

1021 Arnaud Deruelle, Katura Miyazaki, and Kimihiko Motegi, Networking Seifert

surgeries on knots, 2012

1020 Dominic Joyce and Yinan Song, A theory of generalized Donaldson-Thomas

invariants, 2012

1019 Abdelhamid Meziani, On ﬁrst and second order planar elliptic equations with

degeneracies, 2012

1018 Nicola Gigli, Second order analysis on (

(

M), W

), 2012

1017 Zenon Jan Jablo´

nski, Il Bong Jung, and Jan Stochel, Weighted shifts on directed

trees, 2012

1016 Christophe Breuil and Vytautas Paˇ

sk¯

unas, Towards a modulo p Langlands

correspondence for GL

, 2012

1015 Jun Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, 2012

1014 R. Fioresi and F. Gavarini, Chevalley supergroups, 2012

1013 Kaoru Hiraga and Hiroshi Saito, On L-packets for inner forms of SL

, 2012

1012 Guy David and Tatiana Toro, Reifenberg parameterizations for sets with holes, 2012

1011 Nathan Broomhead, Dimer models and Calabi-Yau algebras, 2012

1010 Greg Kuperberg and Nik Weaver, A von Neumann algebra approach to quantum

metrics/Quantum relations, 2012

1009 Tarmo J¨

arvilehto, Jumping numbers of a simple complete ideal in a two-dimensional

regular local ring, 2011

1008 Lee Mosher, Michah Sagee, and Kevin Whyte, Quasi-actions on trees II: Finite

depth Bass-Serre trees, 2011

1007 Steve Hofmann, Guozhen Lu, Dorina Mitrea, Marius Mitrea, and Lixin Yan,

Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaﬀney
estimates, 2011

1006 Theo B¨

uhler, On the algebraic foundations of bounded cohomology, 2011

1005 Frank Duzaar, Giuseppe Mingione, and Klaus Steﬀen, Parabolic systems with

polynomial growth and regularity, 2011

1004 Michael Handel and Lee Mosher, Axes in outer space, 2011

1003 Palle E. T. Jorgensen, Keri A. Kornelson, and Karen L. Shuman, Iterated

function systems, moments, and transformations of inﬁnite matrices, 2011

1002 Man Chun Leung, Supported blow-up and prescribed scalar curvature on S

, 2011

1001 N. P. Strickland, Multicurves and equivariant cohomology, 2011

1000 Toshiyuki Kobayashi and Gen Mano, The Schr¨

odinger model for the minimal

representation of the indeﬁnite orthogonal group O(p, q), 2011

999 Montserrat Casals-Ruiz and Ilya Kazachkov, On systems of equations over free

partially commutative groups, 2011

998 Guillaume Duval, Valuations and diﬀerential Galois groups, 2011

997 Hideki Kosaki, Positive deﬁniteness of functions with applications to operator norm

inequalities, 2011

996 Leonid Positselski, Two kinds of derived categories, Koszul duality, and

comodule-contramodule correspondence, 2011

995 Karen Yeats, Rearranging Dyson-Schwinger equations, 2011

994 David Bourqui,

Fonction zˆ

eta des hauteurs des vari´

et´

es toriques non d´

eploy´

ees, 2011

993 Wilfrid Gangbo, Hwa Kil Kim, and Tommaso Pacini, Diﬀerential forms on

Wasserstein space and inﬁnite-dimensional Hamiltonian systems, 2011

992 Ralph Greenberg, Iwasawa theory, projective modules, and modular representations,

2011

991 Camillo De Lellis and Emanuele Nunzio Spadaro, Q-valued functions revisited, 2011

SELECTED TITLES IN THIS SERIES

990 Martin C. Olsson, Towards non-abelian p-adic Hodge theory in the good reduction case,

2011

989 Simon N. Chandler-Wilde and Marko Lindner, Limit operators, collective

compactness, and the spectral theory of inﬁnite matrices, 2011

988 R. Lawther and D. M. Testerman, Centres of centralizers of unipotent elements in

simple algebraic groups, 2011

987 Mike Prest, Deﬁnable additive categories: Purity and model theory, 2011

986 Michael Aschbacher, The generalized ﬁtting subsystem of a fusion system, 2011

985 Daniel Allcock, James A. Carlson, and Domingo Toledo, The moduli space of

cubic threefolds as a ball quotient, 2011

984 Kang-Tae Kim, Norman Levenberg, and Hiroshi Yamaguchi, Robin functions for

complex manifolds and applications, 2011

983 Mark Walsh, Metrics of positive scalar curvature and generalised Morse functions, part I,

2011

982 Kenneth R. Davidson and Elias G. Katsoulis, Operator algebras for multivariable

dynamics, 2011

981 Dillon Mayhew, Gordon Royle, and Geoﬀ Whittle, The internally 4-connected

binary matroids with no M(K

3,3

)-Minor, 2010

980 Liviu I. Nicolaescu, Tame ﬂows, 2010

979 Jan J. Dijkstra and Jan van Mill, Erd˝

os space and homeomorphism groups of

manifolds, 2010

978 Gilles Pisier, Complex interpolation between Hilbert, Banach and operator spaces, 2010

977 Thomas Lam, Luc Lapointe, Jennifer Morse, and Mark Shimozono, Aﬃne

insertion and Pieri rules for the aﬃne Grassmannian, 2010

976 Alfonso Castro and V´

ıctor Padr´

on, Classiﬁcation of radial solutions arising in the

study of thermal structures with thermal equilibrium or no ﬂux at the boundary, 2010

975 Javier Rib´

on, Topological classiﬁcation of families of diﬀeomorphisms without small

divisors, 2010

974 Pascal Lef`

evre, Daniel Li, Herv´

e Queﬀ´

elec, and Luis Rodr´

ıguez-Piazza,

Composition operators on Hardy-Orlicz space, 2010

973 Peter O’Sullivan, The generalised Jacobson-Morosov theorem, 2010

972 Patrick Iglesias-Zemmour, The moment maps in diﬀeology, 2010

971 Mark D. Hamilton, Locally toric manifolds and singular Bohr-Sommerfeld leaves, 2010

970 Klaus Thomsen, C

∗

-algebras of homoclinic and heteroclinic structure in expansive

dynamics, 2010

969 Makoto Sakai, Small modiﬁcations of quadrature domains, 2010

968 L. Nguyen Van Th´

e, Structural Ramsey theory of metric spaces and topological

dynamics of isometry groups, 2010

967 Zeng Lian and Kening Lu, Lyapunov exponents and invariant manifolds for random

dynamical systems in a Banach space, 2010

966 H. G. Dales, A. T.-M. Lau, and D. Strauss, Banach algebras on semigroups and on

their compactiﬁcations, 2010

965 Michael Lacey and Xiaochun Li, On a conjecture of E. M. Stein on the Hilbert

transform on vector ﬁelds, 2010

964 Gelu Popescu, Operator theory on noncommutative domains, 2010

963 Huaxin Lin, Approximate homotopy of homomorphisms from C(X) into a simple

∗

-algebra, 2010

For a complete list of titles in this series, visit the

AMS Bookstore at www.ams.org/bookstore/.

ISBN 978-0-8218-5312-2

9 780821 853122

MEMO/217/1019

Document Outline

Introduction
Chapter 1. Preliminaries
Chapter 2. Basic Solutions
Chapter 3. Example
Chapter 4. Asymptotic behavior of the basic solutions of L
Chapter 5. The kernels
Chapter 6. The homogeneous equation L u=0
Chapter 7. The nonhomogeneous equation L u=F
Chapter 8. The semilinear equation
Chapter 9. The second order equation: Reduction
Chapter 10. The homogeneous equation Pu=0
Chapter 11. The nonhomogeneous equation Pu=F
Chapter 12. Normalization of a Class of Second Order Equations with a Singularity
Bibliography

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