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AFOSR-1 01 7

NEW YORK UNIVERSITY

INSTITUTE OF MATHEMATICAL SCIENCES

LIBRARY

4 WMhrnjIon Plate, New York 3, N. Y.

^^E ET PR/1^ NEW YORK UNIVERSITY

Su I In TjL ^ Institute of Mathematical Sciences
'V^ y^ Division of Electromagnetic Research

RESEARCH REPORT No. BR-33

Some Uniqueness Theorems for the
Reduced Wave Equation

LEO M. LEVINE

Mathematics Division

Air Force Office of Scientific Research

Contract No. AF4 9(6 3 8)-229

Project No. 47500

JUNE 1961

aFOSR- 1017

NEW YORK UNIVERSITY
Institute of Mathematical Sciences
Division of Electromagnetic Research

Research Report No. BR- 55

SOME UNIQUENESS THEOREMS FOR THE REDUCED WAVE EQUATION

Leo M. Levine

o^ yyi ,5^^ - >-e_.

Leo M. Levine

%vr/. y^,

June 1961

fK£^

Morris KLlne
Director

Qualified requestors may obtain copies of this report from the
Armed Services Technical Information Agency^ Arlington Ball
Station, Arlington 12, Virginia. Department of Defense con-
tractors must be established for ASTIA services, or have their
'need to know' certified by the cognizant military agency of
their project. Reproduction in whole or in part is permitted
for any purpose of the United States Government.

Abstract

This paper deals with various extensions of the Magnus-Rellich uniqueness
theorem for the reduced wave equation in infinite domains. The theorem is ex-
tended to cover piecewise smooth boundary surfaces of a general kind, and mixed
boundary conditions; no auxiliary "edge conditions" are required at edges or at
discontinuities in the boundary conditions - continuity of the wave function in
the closure of the domain is sufficient. Another extension treats infinite
boundaries; for real values of the propagation constant, these are restricted to
surfaces which are (generalized) cones sufficiently far from the origin.

Introduction

Page

Definitions; Statement of the Uniqueness Theorem for the

Exterior of a Regular Closed Surface 6

On the Behavior of Solutions Near Regular Boundary Points 11

k. On the Behavior of Solutions Near Edges 35

5. Proof of the Uniqueness Theorem for the Exterior of a Regular

Closed Surface 37

6. Extension to a More General Class of Finite Boundaries kO

7. Extension to Infinite Boundaries and to Boundaries with

Conical Points 1|.3

Supplementary Results 50

Appendix I: Regularity of Continuous Solutions of the

Reduced Wave Equation 59

Appendix II: A Property of Regular Closed Surfaces. 79

Appendix III: The Eigenvalue Problem for the Beltrami Operator

in Spherical Surface Regions. 78

References ^^.

- 1 -

1. Introduction.

The purpose of this paper is to complete and extend in a number of ways the
Magnus-Rellich uniqueness theorem for the reduced wave equation in infinite domains,
The theorem was stated by Rellich L -' essentially as follows:

Let G be the exterior of a finite closed surface B. There exists at most one
function u = u(x ,x , . . . ,x^) defined in G = GuB such that

(a) u e C^ -^ in G

(b) Au + k u =

(k > 0)

(c) u assumes given values on I

n-1

(d) lim r ^ (Su/Sr - iku) =

r-Ka

, 2 2

*^b

1/2

uniformly along all rays from the origin.

(e) B and u satisfy such regularity conditions as to ensure the validity of
the following case of Green's formula:

— 2 — 2 —
w(Aw + k w) - w(A w + k w)

dV = / (ww - WW )dS
J I" r
r=p

(1.1)

Here w = u-v, where v is any other solution which satisfies the same regularity
conditions as u, as well as condition (d), and assumes the same values on B.
G is the domain exterior to B and interior to the hypersphere with center at
the origin and radius p. The radius p is so large that the hypersphere con-
tains B in its interior.

We shall show, in the three dimensional case, that (e) may be replaced by the
simple conditions that u be a solution of the wave equation (b) which is continuous
in G, and that B belong to a certain fairly general class of piecewise smooth
closed surfaces (herein designated "regular closed surfaces".) No auxiliary

"edge conditions" will be required of u at edges or vertices. Moreover^ we
shall extend the theorem to include the mixed boundary value problem; i.e.,
instead of condition (c), u may be specified on some parts of B, and Su/Sn + 3u
specified over the remainder of B, p being in general a (possibly discontinu-
ous) function of position on B.

The remaining conditions of the above theorem will also be generalized
or deleted. It will be proved that condition (a) is superfluous, being de-
ducible from the continuity of u and condition (b). In condition (b) we shall
allow k to be any complex niomber other than zero satisfying Im k g o, Re k ^ 0.
The radiation condition (d) will be replaced by the weaker integral condition

r I IP

li^ / |Su/dr - lku| dS = 0. (1.2)

r=p

The resulting statement. Theorem 2.1 below, is still not general enough to
cover a number of problems of interest in diffraction theory, for example, the
problem of diffraction by a semi-infinite circular cone. Besides being infinite,
this surface possesses a singularity - its apex - which is not permitted by the
hypothesis of Theorem 2.1. Moreover, the restriction to closed surfaces rules
out the application of the theorem to such situations as diffraction by a disk
of zero thickness. The latter restriction is not hard to remove, however, and
the theorem will first be extended to admit a considerably more general class of
finite boundaries . It will then be further extended to include Infinite boiond-
aries, and boundaries with singularities of the type occurring at the apex of a
(generalized) cone. When k is a real number, the extension to infinite boundaries
is limited to the case where the domain is "eventually conical", i.e. outside
a sufficiently large sphere, it is generated by a ray emanating from the center.

- 5 -

Furthermore, it will be ass\;med in this case that sufficiently far from the center
only the Dirichlet or Neumann boundary conditions obtain, possibly changing from
one to the other along a generatrix of the cone. In the case of domains with in-
finite boundaries, the integral form of the radiation condition (1.2) will also
be used, the region of integration being that part of the sphere of radius p con-
tained in the domain.

The original uniqueness theorem for the reduced wave equation in infinite
domains was stated by Sommerf eld ^ J ^ along with the radiation condition an

u = 0(^ )' ^^ ^ — ^ °° (1-5)

uniformly with respect to direction. The theorem was first proved in this form
by Magnus ^ -I . The elimination of the extraneous condition (1.5) was accomplished
shortly afterwards by Rellich, resulting in the theorem stated above. Magnus dealt
with the Dirichlet problem for smooth closed boundaries, and assumed the existence
and continuity of ^/Sn on the boundary. As seen above, Rellich also restricted
himself to the Dirichlet problem, but did not give explicit conditions for the
boundary surface, or for the behavior of the wave function at the boundary. In-
stead, he gave the complicated implicit condition (e). Both authors assumed k
to be real. This restriction was removed by several writers, of whom Atkinson "- -"
appears to have been the first.

Diffraction at edges has been discussed by a number of writers, among whom
are Melxner '- -• , Bouwkamp L' J and Peters and Stoker '■ -' ; in every case auxiliary
edge conditions were assumed. Meixner proved uniqueness for the electromagnetic
problem using a finite energy condition, as well as the assumption that the
field could be expanded In the vicinity of the edge in a series of positive

- k -

fractional powers of r (\iiere r is the distance to the edg^. He stated^ without
proof or references, that in the acoustic case, boundedness of the solution is
a sufficient condition for uniqueness. Bouwkamp considered diffraction by an
aperture in an acoustically hard or soft screen^ he assumed boundedness and a
finite energy condition in the vicinity of an edge, conjecturing that bounded-
ness alone might be sufficient. Peters and Stoker (who considered the two di-
mensional problem) assumed a condition at a corner of the form

tw| =0(i-"°'). as r — > ; < a < 1. (l.^)

where r is the distance to the corner. Wilcox l -^ apparently proved uniqueness
for the case where the boundary surface is "regular" in the sense of Kellogg L J,
having edges and vertices, but he effectively ruled out such singularities of the
surface by requiring that the solution be of class C (G;

The general mixed boundary value problem in diffraction theory has apparently
not been considered elsewhere, although some special problems have been investi-
gated. The third boiindary value problem (for smooth surfaces) was treated by

T • [12]

Leis .

Regarding infinite boundaries, Peters and Stoker ^ -^ proved uniqueness for
the case of a boundary in two dimensions which consists, eventually, of two straight
lines extending to infinity. For the case of real k, the extension of the unique-
ness theorem to infinite boundaries presented here treats the three dimensional
analogues of the infinite boundaries of [8], namely, eventually conical boixndaries.
However, in the three dimensional case, where the cones have arbitrary shapes
(e.g., they may have edges extending to infinity) and where mixed boundary condi-
tions are considered, the proof, although along lines similar to that of [8],
involves several interesting points not encountered in the two dimensional
problem Finally, it should be mentioned that Rellich •- -• discussed certain in-
finite domains in which the solution is not unique.

- 5 -

The generality of the results presented here is made posslhle in large mea-
sure by certain estimates "up to the boundary" for the derivatives of solutions

ri5i

of second order elliptic equations. These estimates are due to Schauder ^ -> for
the Dirichlet problem, and to Miranda L J and Agmon, Douglis and Nirenberg L ^
for general boundary conditions .

The plan of the paper is as follows. In Section 2 we define some terms and
give a complete statement of the uniqueness theorem for the exterior of a regular
closed surface. In Sections 5 and k we deal with the behavior near the boundary
of solutions of the wave equation. Essentially, the purpose is to enable us to
derive a formula, (Lemma 5. l), similar to (l.l) for solutions u which satisfy
homogeneous (mixed) boundary conditions on a regular closed surface. In Section
5 we first establish a certain degree of smoothness up to the boundary of u at
regular boundary points, using the limited assumptions of the uniqueness theorem.
This then allows us to use the "Schauder boundary estimates" (we shall use this
designation regardless of which boundary condition is involved) to obtain estimates
for |Vu| near regular boundary points. In Section 4 we use these results to prove
Theorem 4.1, which concerns the behavior of u near edges or discontinuities in the
boundary conditions; essentially, it furnishes an estims.te for |Vu | something like
(1.4). It is this theorem which makes it possible to dispense with accessory edge
conditions. In Section 5 we prove the uniqueness theorem for the exterior of a
regular closed sirrface. In Section 6 the extension to a more general class of
domains with finite boundaries is carried out. Section 7 Is concerned with the
extension to infinite boundaries and to boundaries with conical singularities. In
Section 8 we discuss several supplementary topics. These include the behavior for

* Lemma 5.I corresponds to the first (asymmetric) form of Green's theorem,
rather than the second form used to obtain (l.l); the first form is more
convenient when dealing with complex k.

- 6 -
large r of a wave function about an eventually conical obstacle, and the invari-
ance of the radiation condition under a change of origin. Appendix I consists of
a proof of the statement made above in connection with condition (a) of Rellich's

theorem, namely, that a solution of the waye equation which is continuous in a

(2)

domain G must belong to C (g). Appendix II contains a geometrical result

used in the proof of Theorem h.l. Appendix III contains a proof of the

existence of a complete set of eigenfunctions for the Beltrami operator in

general spherical surface regions (with mixed boundary conditions). In addition,

some required properties of the eigenfunctions and associated eigenvalues are

derived. These results are used in Sections 7 and 8.

2 . Definitions ; Statement of the Uniqueness Theorem for Domains Bounded by
Regiilar Closed Surfaces.

* n

Definition 1. Let R be a region of euclidean n- space, E . A complex- valued

function u on R is In C^™'^(R) for integral m s o, if

th k

(a) For S k ^ m, the k order partial derivatives, D u, exist and are

o ** o
continuous In R, D u = u being continuous in R.

o -^

(b) For X e R - R, llm D u exists, o ^ k S m.

x-*x
o

X€R

f \
It is easily seen that for u € C (R), if the limiting values are assigned to

k °

D u, k ^ m, on R - R, we obtain continuous extensions of the derivatives to all of R.

Definition 2. Let A be a subset of K . A complex-valued function u on A is said
to satisfy a Holder condition with exponent X, < X < 1, If

* We employ the more or less standard usage: a connected open set in E Is a

domain ; a region is a domain plus some (possibly all, or none) of Its boundary points.

_ o
*♦ A, A, and A denote the boundary, closure and interior of A, respectively.

sup

lu{x ) - u(y) | ^^ ^2.1)

x.yeA |x-y|
This least upper bDuad is called the "Holder constant', u is said to be Holder con-
tinuous with exponent X in a region R if it satisfies a Holder condition in every
compact subset of R. For integral m g q and < X < 1, C (r) will denote the
subclass of functions in C (R) whose derivatives of order m, when continuously
extended to R, are Holder continuous with exponent X in R.

Definition 3- AC surface element, (integral m s 1, ? \ < i) is a set of
points in E' which for some system of cartesian coordinates x ,x^,x , admits a
representation

x^ = f(x^,X2) (x^.x^) € D (2.2)

where f e C (D) • Here D is a domain in E" such that D Is a rectifiable Jordan
curve. In addition we require that it be possible to extend f to a function f de-
fined in a domain D containing D where f e C (d). A point of the C surface
element whose projection in the x - x plane lies on D is an edge point . The other

points are inner points of the surface element. If F is a C surface element

o
then F denotes the set of its inner points.

Now let us suppose that y, ,yp,y, constitute another cartesian coordinate sys-
tem where the angle between the y - axis and the tangent plane at any point of the
surface element (2.2) Is bounded away from zero. Then it can easily be shown that
the surface element is also represented by the equation y = g(y-,,y^) where
g e C (D-,)? and D is the projection of the surface element on the y - y^ plane;
g and D satisfy the same boundary and extensibility requirements that f and D
satisfy.

Next, let F be the C^"" ' surface element represented by (2.2) and let
X , S X < 1, and the Integer m satisfy

* We shall often write " u e c'™"*" •'(fi) " without specifying X, meaning that this is
true for some X, < X < 1. We shall use a similar convention in connection with
Definition 3-

^ m + \^ m + X (2.5)

1 1

(m^+X^)

Then C„ (f).

"(m +X ) ■

, will be the set of functions (|)(x) on F, C^ D ^
m^+X^ )

It is not difficult to

(m +X )
where (JjCx^.x^, fCx^.x^)) is in C (D), C " " (d)

(m^+X^)
show that membership of (j> in C (f) is unaffected by a cartesian coordinate

transformation, as long as F has a representation of the type (2.2) in the new co-
ordinate system. (This would not necessarily be true if (2.3) did not hold.)

Definition k. A subset B of E is a closed surface if it is a connected, compact

2-manifold; i.e., every point of B has a neighborhood whose intersection with B is

a 2-cell (a homeoiBorph of the interior of a circle).

5 3

A closed surface B in E separates the set E - B into two components, one of

which is unbounded and called the exterior of B. The bounded component is the

interior of B. As here defined, a closed surface is always homeomorphic to a

sphere with a finite number of handles.

Definition ^. A regular closed surface is a closed surface which can be subdivided
into a finite number of C surface elements satisfying the following condition:
two adjacent surface elements must not form a " zero exterior angle" ; i.e., there
shall be no points of the exterior region inside a cusp formed by two surface
elements tangent at a common edge point.

* [17] p. 526, ll+l.

** If this " cusp condition" were omitted, and C surface elements were re-
placed by "regular surface elements" (Kellogg '- -'), the regular closed surfaces de-
fined here would be equivalent to those of Kellogg. (The defining functions for
regular surface elements, corresponding to f(x^,Xp) in (2.2) satisfy less stringent
smoothness conditions than those for C surface elements, having to be only con-
tinuously differentiablej on the other hand, the boundary cirrves of regular surface
elements are smoother, consisting of a finite number of continuously differentiable
arcs. )

- 9 -

The open sphere of radius p and center x will be denoted by a(p,x); its

boundary by E(p,x).

If R is a region in E , and x e R, then t (p,x) is that component of

a(p,x)nR which contains x.

-^^ will denote the outgoing normal derivative at a boundary point of a
on " "

region in which u is defined. More precisely, suppose that u is defined in a

o ^

region R, x e R - R, and suppose that in a neighborhood of x, R is a surface

possessing a tangent plane at x; then Su/Sn is defined as

lim ^^^^ \ — L if this limit exists, where I is the straight line normal

^ 1^ I n

x-*x I X - X I

xet nR

n

to R at X. (This limit certainly exists if, as in the hypothesis on some parts of

*
the boundary in the follt5wing theorem, lim Vu exists , u being defined and continu-

x-«
o

X€l

ous at x) .

We are now In a position to state the main theorem of this paper, namely:

Theorem 2.1 . Let G be the exterior of a regular closed surface B, and suppose that
u(x), defined in G, satisfies the following conditions:

(a) u € C^°^(G);

(b) u is a solution in G of the reduced wave equation

Au + k\ =0, k ^

(2.4)
Re k, Im k ^ 0;

In this expression it will always be Implicitly presupposed that Vu is

defined in a(p,x)nRfor some p > 0.

2 2
Here we assume only that the three derivatives '^ u/bx. , i=l,2,3, exist.

- 10 -

(c) B has a decomposition into a finite number of C surface elements F
which are divided into two classes jcJ and /-fas follows:

On F inxj , u satisfies the homogeneous Dirichlet boundary condition u = 0.

At every inner point x of F xrij^ , lim Vu exists^ and u satisfies the

x->-x
xcG

boundary condition Su/Sn + 3u = , where P is a non-negative function
belonging to cJ' "*" (f);

lim

p-»oo

Su

^ dS = 0. (2.5)

Then, for all x e G

u(x) = 0. (2.6)

Next, in condition (c), let us replace the homogeneous botmdary conditions
u = 0, and ^/Sn + 0u = by the corresponding inhomogeneous boundary conditions
u = IJJ and ^/Sn + 3u = (j) , respectively, where (J) and ijj are arbitrary functions on
their respective surface elements. Let us call the resulting condition (c' ).
Then it is easy to see that if two functions satisfy conditions (a), (b), (c')
and (d), their difference satisfies (a), (b), (c), and (d), and hence vanishes.
(To show that the difference function satisfies (d) we simply use the triangle

inequality for the norm | v| =/ / |v| dSj .)Thus Theorem 2.1 is equivalent

\^(P.o) J
to the uniqueness theorem that there exists at most one function u satisfying

conditions (a), (b), (c') and (d).

It should be pointed out that the radiation condition (d) is invariant under a

translation of the center of the spheres Z(p, o) from the origin to any finite point.

This will be shown in section 8.

* One of these classes may, of course, be empty.

- 11 -

5- On the behavior of solutions near regular boundary points.

Later, (in the proof of Lemma 5-1)^ it will be necessary to apply Green's

_ — *

theorem to the wave functions u and u in certain subregions of G . These sub-

regions are obtained from G by excluding the edge points ^ of B by means of
"tubular" surfaces arbitrarily close to g. However, the explicit smoothness
conditions on u stated in the hypothesis of Theorem 2.1 are insufficient for
this application of Green's theorem, and we must therefore prove additional smooth-
ness in the interior and at regular boixndary points of G for this purpose.
Moreover, as was indicated at the end of Section 1, additional smoothness
properties are prerequisite for the application of the "Schauder boundary
estimates" ,

Smoothness in the interior is covered by Appendix I, where it is proved that

(2)

the continuity of a solution u of {2.k) in the domain G implies that u e C (g),

(and hence that u is analytic in G *- -' ) . With regard to regular boundary points,
the property we need is that u € C^ (GU F) for each C^ ^ surface element F
inj^U >^(i-e. u e C^ "*" "^(G -^)). The proof is different for the two cases
F € ,/ and F ejy. For F e ^ we begin by stating (without proof) a similar result
for harmonic functions, and from it we derive the desired result for wave functions,

* The domains G are defined in condition (e) of Rellich's theorem in Section 1.
P

** Throughout the paper 5 will denote the set of edge points of the C^ surface

elements into which B is subdivided according to hypothesis (c) of Theorem 2.1.

Thus g includes the points at which there are discontinuities associated with the

boundary conditions as well as the points of the actual edges of B.

*** Smoothness up to the boundary of solutions of general elliptic systems has been

ri9i

proved by Nirenberg ' ' but the general results presented there require more smooth-
ness conditions for the surface element than are required for our special case.

- 12 -

Theorem 3-1 (Kellogg L J) Let G be a domain in E ; let F be a c'"^^ surface

o ^

element contained in 3 such that every point of F has a neighborhood (in E ) which

does not intersect G - F. Suppose that u is harmonic in G and continuous in Gi^,

and that the boundary values of u on F are of class C„^™ (F), where m = 1 or 2,

< X' < 1, and m+X ' g 2 + X. Then u e C^"^"^^ ' ^(GiJF) •*

[211
Lemma 3 • 1 (Korn '- -' ) Let f (x) be a bounded integrable function on a bounded

domain G in E and define

^(^) = f 1~^ ^V- (5.1)

Jx-x'
G

Then w e C^ "*" ^(g) for any X such that < X < 1. If, in addition, f(x) 6 c'^^(g),
then w e C^^^^^G).**

Theorem 3-2 Let G and F be as in Theorem 3.1; suppose that u is in C (GUF)>

satisfies the wave equation {2.k) in G, and vanishes on F. Then u e C (GIJF).

Proof : For any x e F, the assiunptions on F and G imply that there is a

p = p (x) > such that the region t (p ,k) possesses the following

'^ ° GUF °

properties :

* This theorem is proved in [ll] for any m = 1,2, 3^ ...^ provided that if m > 2,
F is a C surface element. It should be pointed out that in Kellogg 's statement
of the theorem there are additional conditions on the surface element which are not
satisfied by our C surface elements F; however, for every inner point x of F
we can find a sufficiently small piece of F containing x which does satisfy the
additional conditions, whence it is easy to show that the theorem follows as stated.
See also Graves '- J, where a gap in Kellogg 's proof is filled.

** The statement obtained by replacing G by G throughout the lemma is what is act-
ually proved in [2l] ; moreover, there G is assumed to be the interior of a regular
closed surface without edges (i.e. for every x e G there is a subdivision of G into
C surface elements F such that x is an inner point of some F.) However, upon
applying the results of [2l] to arbitrary spheres whose closures are in G, we easily
obtain the lemma as stated.

- 13 -

a) T o(p,^) = a{p^,^n (GUF);
GuF °

o
t) T q( p ,5) is contained in GUF, and consists of a region on the surface

GUF
E(p ,x) and a C^ "*" surface element F^CF.

We shall prove that u € C^'""*' (t) for any region t = t ^(p ^x);, from which the

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