\documentclass{amsart}
\begin{document}
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.\ 1999(1999), No.~20, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 1999 Southwest Texas State University and
University of North Texas.}
\vspace{1cm}
\title[\hfilneg EJDE--1999/20\hfil On a generalized reflection law]
{On a generalized reflection law for functions satisfying the Helmholtz equation}
\author[Dawit Aberra\hfil EJDE--1999/20\hfilneg]
{Dawit Aberra}
\address{Dawit Aberra \hfill\break
Department of Mathematical Sciences,
University of Arkansas \hfill\break
Fayetteville, AR 72701, USA\hfill\break
E-mail address: daberra@comp.uark.edu \hfill\break
http://comp.uark.edu/$\sim$daberra}
\date{}
\thanks{Submitted October 9, 1998. Published June 4, 1999.}
\subjclass{31A05, 31A35, 31B05 }
\keywords{Reflection law, Helmholtz operator}
\begin{abstract}
We investigate a generalized point to point reflection law for the solutions of
the Helmholtz equation in two independent variables, obtaining results that
include some previously known results of Khavinson and Shapiro as special cases.
As a consequence, we obtain partial negative answers to the ``point to compact
set reflection'' conjecture suggested by Garabedian and others.
\end{abstract}
\maketitle
\newtheorem{theorem}{Theorem}[section] % theorems numbered with section #
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}
\section{ Introduction }
According to a result of H. Lewi \cite{lewi}, if $u$ is a function of two
variables and satisfies a partial differential equation with the Laplacian
in the principal part in a domain $D$ adjacent to the real axis
$\mathbb{R}$ and $u_{|_\mathbb{R}} =0$,
then $u$ extends to a mirror image $D^\prime $ of $D$ with respect to
$\mathbb{R}$.
However, with regard to point to point ``reflection laws'', the situation
for operators
slightly different from the Laplacian (or the wave operator) is
drastically different. The following theorem is due to
Khavinson and Shapiro, \cite{khavinsons} (see also \cite{khavinson}, and
compare with
Study's interpretation of the Schwarz reflection principle).
\begin{theorem} \label{P:0.1}
Let $\gamma = \{ (s, t) \mid \;t=g(s) \}$, with $g'(s)
\neq 0$, be a non-singular real analytic curve in $\mathbb{R}^2$. If for
two points
$P$ and $Q$ sufficiently close to $\gamma $ (but off
$\gamma )$ there exist a
constant $c = c(P, Q)$ such
that the ``Schwarz reflection law''
\begin{equation} \label{E:0.1}
u(P) + c\,u(Q)=0
\end{equation}
holds $ \forall u, u_{| _\gamma }=0 $ and satisfying the ``Helmholtz
equation''
\begin{equation} \label{E:0.2}
\frac {\partial^2 u}{\partial s \partial t} +\lambda ^2u = 0,
\end{equation}
where $\lambda > 0$, then $c =1$, $P$ and $Q$ must be symmetric with
respect to $\gamma $ and $\gamma $ must be a straight line.
\end{theorem}
By using Study's change of variables $z= X+iY,\, w=X-iY$ that reduces the
``real'' Helmholtz operator $\triangle +\lambda^2 $ to the complex
hyperbolic
operator $4\frac{\partial ^2}{\partial z\partial w} + \lambda^2$, a
similar conclusion as in Theorem \ref{P:0.1} holds $\forall u,
u{|_\gamma} = 0$ and satisfying $\triangle u +\lambda^2u =0$.
Suppose $\Gamma $ is a non-singular real analytic
hypersurface defined in some domain $\Omega \subseteq \mathbb{R}^n$,
and $L$ is a partial differential operator in $\mathbb{R}^n$. Theorem
\ref{P:0.1} above can be considered as a partial
answer to the first of
the following two general problems:
$(1)$ For which pair of points $P\neq Q$ in $\Omega \setminus \Gamma $ is
it
true that there is a constant $c=c(P, Q)$ such that
\begin{equation}
u(P) + cu(Q)=0
\end{equation}
for all solutions of $Lu=0$ in $\Omega $, vanishing on $\Gamma $ ?
We say the pair $P, Q$ has the \emph{(point to point) reflection property}
(with respect to $\Omega , \Gamma \text{ and }L )$ if there is such a
constant $c$.
$(2)$ (In general when the point to point reflection property fails ):
Does a ``point to compact set reflection'' hold, i.e., given a point
$P_0$,
is there a compact set $K=K(P_0, \Gamma , L) \subseteq \Omega $ (on the
``other side '' of $\Gamma )$ and a measure (or distribution)
$T = T(P_0, \Gamma , L)$ supported on $K$ such that
$$
u(P_0 ) = \langle T, u\rangle \text{ for all solutions of } Lu=0 \text{
vanishing on } \Gamma \,\,?
$$
Problems of type $(1)$ have been studied by several
researchers (\cite{lewi}, \cite{G},
\cite{khavinsons}, \cite{ek}, \cite{ebenfelt}; see also
references there). This problem was the theme of the paper \cite{ek} in
the special case
$L=\Delta $ (the Laplace operator) and is completely settled there for
any dimension $n\geq 3$. There, it was shown that such pairs of points
are
very rare in $\mathbb{R}^n $ when $n$ is even (and $>3$), and
that the
reflection
property never holds when $n$ is odd (and $\geq 3$), unless $\Gamma $
is a
sphere or a hyperplane, in contrast to the case $n=2$ and the classical
Schwarz
reflection principle. When $L$ is the Helmholtz operator and $n=2$, no
pair of points have the reflection property unless $\Gamma $ is a
line (this is Theorem \ref{P:0.1} above, which is an implicit result in
\cite{khavinsons} ).
The case of the real Helmholtz operator for $n=3$ was studied in
\cite{ebenfelt}, a striking result there (among others) being,
in contrast to the $n=2$ case, there are hypersurfaces $\Gamma $ other
than hyperplanes and pairs of points where the reflection property holds.
So far, no result related to the ``point to compact set reflection''
problem was published, although it was conjectured by Garabedian, and
later by
the authors of the papers \cite{khavinsons} and \cite{ek},
that it is likely to get positive results in the instances of
failure of the
point to point reflection property. An explicit calculation supporting
this conjecture for the (real) Helmholtz operator in two dimensions has
been done
in \cite{sss} (or see their rather widely
available book, \cite{ss}). More precisely, if $\Omega $ is
a domain in
$\mathbb{R}^2$
divided by a non-singular real analytic curve $\gamma $ into two parts, a
formula which gives the value at a point $P_0\in
\Omega \setminus \gamma $ of a
function $u$ satisfying the (real) Helmholtz equation near $\gamma $ and
vanishing on $\gamma $ is given in terms of an integral involving $u$ and
its
first order derivatives, evaluated along a path joining $\gamma $ to
the
point symmetric to $P_0$ with respect to $\gamma $.
Our main result in this paper is Theorem \ref{T:2.1},
which says that under the same hypothesis as in Theorem \ref{P:0.1}, if
the ``Schwarz reflection law'' given by \eqref{E:0.1} is replaced by the
weaker hypothesis,
\begin{equation}\label{E:100}
u(P) = \sum_{| \alpha | \leq N }c_\alpha \, D^\alpha
u (Q)\;,
\end{equation}
where $N$, $c_\alpha $ $( | \alpha | \leq N) $ are constants
depending
only
on $P$ and $Q$, then the
conclusion of the theorem remains the same. In other words, this means
that the reflection property in the case of the two
dimensional Helmholtz operator, which was known to fail
for the ``Schwarz reflection law'' given by \eqref{E:0.1}
(when $\gamma $ is not a line),
also fails even when this
law is replaced by the weaker hypothesis given by \eqref{E:100}, which we
may refer to as the ``generalized (Schwarz) reflection law''. This entails
similar failures in the case of the two dimensional ``real'' Helmholtz
operator (when $\Gamma $ is not a line) and in the case of the
three dimensional Laplace operator when $\Gamma $ is a non-planar
cylinder, see Corollaries \ref{C:9.3}\,(i), \ref{C:8.3}\,(i). All these
results generalize previously known
results in \cite{khavinsons}. As a consequence, related partial negative
answers to the `point to compact set
reflection' conjecture mentioned above are obtained, i.e., in the case of
the two dimensional (real) Helmholtz operator (when $\gamma $ is not a
line) and in the case of the three dimensional Laplace operator (when
$\Gamma $ is a non-planar cylinder), there is no
point to point reflection in the sense of problem (2) mentioned above, see
Corollaries \ref{C:8.1}, \ref{C:9.3}\,(ii) and \ref{C:8.3}\,(ii).
\section{Preliminaries}
If
$$
L= \frac{\partial^2}{\partial s \partial t } +a(s, t)
\frac{\partial
}{\partial s } +b(s, t)\frac{ \partial}{\partial t} +c(s, t)
$$
is a hyperbolic differential operator, where $a$, $b$ and $ c $ are entire
functions of two variables, its adjoint is defined by
$$
L^\star u =\frac{ \partial ^2u}{\partial s \partial t}-\frac{\partial
}{\partial s }(au) -\frac{\partial }{\partial t } (bu) +cu.
$$
The Riemann function $R_L :=R(s,t;x,y)$ at a point $(x, y)$
for the operator $L$ is defined as the solution of the following
Cauchy-Goursat problem
$$\begin{gathered}
L^\star R = 0 \text{ near } (x, y)\,,\\
R(x, t; x, y) = \exp{ \int_y^t a(x, \tau )\,d\tau }\,,\\
R(s,y; x,y) = \exp{ \int_x^s b(\tau , y)\,d\tau }\,.
\end{gathered}$$
If we define $r(s)=R(s, y; x, y) $ and $s(t)= R(x, t; x, y), $ it is easy
to see that $r_s-br=0$ on $\{t=y\},$ $ s_t -as=0$ on
$\{ s=x\} $ and $R(x, y; x; y) =1$. Moreover, it is known that $R(s, t;
x,y)$
is an entire function of all four variables and if $L$ is the
Helmholtz
operator, then
$$
R(s, t; x, y) = J_0 (2\lambda \sqrt{(s-x)(t-y)}\,)\,,
$$
where $J_0$ is the zero Bessel function. For these and other properties of
the Riemann function, we refer to \cite{hadamard}.
Let $\gamma $ be as in Theorem \ref{P:0.1}, $P=P(x, y)$ be a point
sufficiently close to $\gamma$. Let
\begin{eqnarray*}
&A_P =(g^{-1}(y), y),\; B_P = (x, g(x)) &\\
&U= aRu +\frac{1}{2} Ru_t -\frac{1}{2} R_tu & \\
&V= bRu + \frac{1}{2} Ru_s -\frac{1}{2}R_su\,. &
\end{eqnarray*}
By a straightforward calculation, we have
\begin{eqnarray*}
0 &=&R(Lu) -u(L^\star R) \\
&=& U_s+V_t \,.
\end{eqnarray*}
If $\mathbf{G}$ denotes the region bounded by segments $PA, PB$ and the
arc
$AB$, applying Green's theorem we obtain
$$\oint_{\partial \mathbf{G}} Udt-Vds =0 \,.$$
Simplifying this using the properties of the Riemann function and using
the fact
that $u_{| _\gamma }=0$, we obtain the following formula of Riemann:
\begin{equation} \label{E:1.1}
u(P) = \frac{1}{2} \int_{A_P}^{B_P} R(\frac{ \partial u}{\partial
s}ds-\frac{\partial u}{\partial t} dt).
\end{equation}
\section{Main results}
\begin{theorem}\label{T:2.1}
Suppose
\begin{equation}\label{E:2.1}
u(P) = \sum_{| \alpha | \leq N }c_\alpha \, D^\alpha
u (Q)
\end{equation}
$\forall u, u_{|_\gamma} =0 $ and satisfying the Helmholtz equation
\eqref{E:0.2}, where $P$ and $Q$ ($P\neq Q$) are points in $\mathbb{R}^2$
sufficiently
close to $\gamma $ (but off $\gamma $), $c_\alpha = c_\alpha (P, Q)$ and
$N=N(P,Q)$ is an integer.
Then, $c_\alpha = 0$ $\forall
\alpha \neq 0$, $c_0 =-1$, $P$ and $Q$ must be symmetric with respect to
$\gamma $ and $\gamma $ must be a straight line.
\end{theorem}
\begin{proof}
Since $u_{xy} =-\lambda^2 u $, there is no loss of generality in assuming
that there are no mixed derivatives involved in the hypothesis \eqref{E:2.1}.
So let
\begin{equation}
\label{E:2.2}
u(P)= \sum_{n=0}^{N} c_n (D_x^n u)(Q) + \sum_{n=1}^N d_n (D_y^n u)(Q)\,,
\end{equation}
where $c_n$, $d_n$ ($0\leq n \leq N $) are constants that depend only on
$P$
and $Q$. If $P$ and $Q$ are sufficiently close to $\gamma $, all the
solutions of the Cauchy problem
\begin{equation}\label{E:C}
\begin{gathered}
\frac{\partial ^2 u}{\partial s \partial t} +\lambda ^2 u = 0\,, \\
\partial ^\alpha u_{|_\gamma }
=\partial ^\alpha ((t-g(s))p(s, t))_{| _\gamma }, \;
| \alpha | \leq 1 \,,
\end{gathered}
\end{equation}
with $p(s, t)$ being a polynomial, are real analytic in a fixed
neighbourhood containing $P$ and $Q $.
For $u$ satisfying the Cauchy data in \eqref{E:C} we have,
$$
(\frac{\partial u}{\partial s} ds - \frac{\partial u}{\partial t}dt
)_{|_\gamma }
= -2pg'(s)_{ | _\gamma }\,.
$$
Using this, \eqref{E:1.1} reduces to
\begin{equation}\label{E:2.4}
u(P)=-\int_{A_P}^{B_P} R(s, t; P)g'pds\,,
\end{equation}
where $p(s, t)$ is a polynomial. Differentiating \eqref{E:2.4}
\emph{n-}times with
respect to $x$, we obtain the following expression for the \emph{n-th}
derivative of $u$ at the point $P$:
\begin{equation}\label{E:2.5}
\begin{aligned}
(D_x^n u) (P) =& - \int_{A_P}^{B_P} (D_{x}^{n} R)(s, t;P)g'pds\\
&- \sum_{k=0}^{n-1} D_{x}^{k} (( D_{x}^{n-1-k} R) (B_P ;
P)g^\prime (x) p(B_P)).
\end{aligned}
\end{equation}
Denote the sum in the right hand side of \eqref{E:2.5} by $v_n$ $(n=1, 2,
\dots , N ) $ and note that we can write,
\begin{equation} \label{E:2.6}
\sum_{n=1}^{N} c_n v_n = \sum_{| \alpha | \leq N-1} M_\alpha
\, (D^\alpha p)(B_P)\;,
\end{equation}
with
\begin{equation}\label{E:2.7}
M_{(n-1, 0)} = c_n R(B_P ; P)g'(x) + \sum_{j=n+1}^{N}c_j
K_{n-1}^{j}, \;\; 1\leq n \leq N\,,
\end{equation}
where $M_{(n-1, 0)}$ denotes the coefficient $M_\alpha $ that appears in
the right hand side of \eqref{E:2.6} for $\alpha = (n-1, 0)\,$ $(1\leq n
\leq N )$ and $K_{n-1}^{j}, $ whose exact values we need not compute,
depend on
$(D_x R)(B_P; P)$, $g^\prime (x) $ and their derivatives. Similarly, we
have
\begin{equation}\label{E:2.77}
\begin{aligned}
(D_y^n u) (P) = &- \int_{A_P}^{B_P} (D_{y}^{n} R)(s, t;P)g'pds\\
&+\sum_{k=0}^{n-1} D_{y}^{k} (( D_{y}^{n-1-k} R)(A_P ;
P)p(A_P)).
\end{aligned}
\end{equation}
Again, note that if we denote the sum in the right hand side of
\eqref{E:2.77}
by $w_n$ $(1\leq n \leq N ), $ we have
\begin{equation}\label{E:2.8}
\sum_{n=1}^{N} d_n w_n = \sum_{| \alpha | \leq N-1} N_\alpha
\, (D^\alpha p)(A_P)\,,
\end{equation}
with
\begin{equation} \label{E:2.9}
N_{(0, n-1)} = d_n R(A_P ; P) + \sum_{j=n+1}^{N}d_j
L_{n-1}^{j}, 1\leq n \leq N\,,
\end{equation}
where $N_{(0, n-1)}$ and $ L_{n-1}^{j} $ are similarly defined.
Replacing $P$ by $Q$ in the differentiation formulae \eqref{E:2.5} and
\eqref{E:2.77}, and using \eqref{E:2.6} and \eqref{E:2.8} in the
hypothesis
\eqref{E:2.2} we obtain
\begin{equation}\label{E:2.10}
\begin{aligned}
\int_{A_P}^{B_P} R(s, t; P)g'pds
- \sum_{n=0}^{N} c_n &\int_{A_Q}^{B_Q} (D_x^n R)(s, t;
Q_)g'p\,ds \\
- \sum_{n=1}^{N} d_n \int_{A_Q}^{B_Q} (D_y^n R) (s, t; Q)
& g'p\,ds\\
& = \sum_{n=1}^{N} c_n v_n - \sum_{n=1}^{N} d_n w_n \\
& = \sum_{| \alpha | \leq N-1} M_\alpha \, (D^\alpha
p)(B_Q) - \sum_{| \alpha | \leq N-1 } N_\alpha \,
(D^\alpha p)(A_Q),
\end{aligned}
\end{equation}
for all polynomials $ p(s, t)$.
Hence,
\begin{equation}\label{E:2.11}
\begin{aligned}
\int_{A_P}^{B_P} R(s,t;P)g'pds =&\int_{A_Q}^{B_Q}R^{\star }(s, t; Q)
g'pds \\
&+ \sum_{| \alpha | \leq N-1 }
M_\alpha (D^\alpha p)(B_Q)
- \sum_{| \alpha | \leq N-1 } N_\alpha (D^\alpha p)(A_Q),
\end{aligned}
\end{equation}
where
$$
R^\star (s, t; Q)= \sum_{n=0}^N c_n(D_x^n R)(s, t; Q) + \sum_{n=1}^N
d_n
(D_y^nR)(s,t;Q)\,,
$$
for all polynomials $p(s, t).$
Suppose $A_P < A_Q $ (with respect to an obvious order on $\gamma $
induced
by parametrization). Without loss of generality, assume $A_Q \leq B_P
\leq B_Q$. Let $T$ be a point on $\gamma $ such
that $A_P < T < A_Q $. Using that $J_0 > 0$ near the origin and
$g' \neq 0$, choose
sequence of polynomials $p(s, t) $ such that
$$
| \int_{A_P}^{T} Rg'p\,ds | \geq \eta \;,
$$
where $\eta $ is a pre-assigned positive number, while for each $\alpha$,
$0\leq | \alpha | \leq N-1$,
$ | D^\alpha p(s, t)| $
can be made arbitrarily small in $(T, B_Q ]$. Using this in
\eqref{E:2.11},
we get a contradiction. Therefore, we must have $A_Q \leq A_P$.
Similarly, $B_P \leq B_Q $.
Hence, \eqref{E:2.10} can be written as
\begin{equation}\label{E:2.12}
\begin{aligned}
\int_{A_Q}^{B_Q} \{R(s, t; P)\chi_{[A_P,B_P ]}
&- \sum_{n=0}^{N} c_n (D_x^n R)(s,t;Q)
-\sum_{n=1}^{N} d_n (D_y^n R)(s, t;Q)\}g'pds \\
& =\sum_{| \alpha | \leq N-1 } M_\alpha (D^\alpha p)(B_Q)-
\sum_{| \alpha | \leq N-1 }N_\alpha (D^\alpha p) (A_Q)\,,
\end{aligned}
\end{equation}
for all polynomials $p(s, t)$.
Let $A_Q =(a, a' )$ and $B_Q = (b, b')$. Consider the sequence of
polynomials,
$$
p_{k} (s, t) ={ \left( \frac{s-a}{b-a} \right)}^k \,, \; \;k \geq N \,.
$$
Observe that for each $k$, $(D_{s}^{j}p_k) (A_Q) =0$,
$(D_{s}^{j}p_k)(B_Q) = 1 $
( where $D_{s}^{j} $ denotes the $j$-th
derivative with
respect to $s$, $0\leq j \leq N-1$), $| p_k(s, t) | < 1 $
in $(A_Q, \; B_Q)$ and that $ | p_k (s, t) | \to 0$ a.e. on $[A_Q,\;
B_Q],$ as $ k \to \infty $. Moreover, since for each $k$, $p_k(s, t) $
does not depend
on $t$ (the second variable), all the derivatives of the polynomials $p_k$
with respect to $t$ are zero.
Using this sequence of polynomials and applying
Lebesgue Dominated Convergence Theorem, we find that the integral in the
left hand side of \eqref{E:2.12} goes to zero, as $k \to \infty $. The right hand
side of \eqref{E:2.12} reduces to
$$
\sum_{n=0}^{N-1} \frac{M_{(n, 0)}}{(b-a)^{n}} \, \frac{k!}{(k-n)!} \,
=: q(k)\,,
$$
where $q(x) $ is a polynomial (of degree $\leq N-1)$. Since the left hand
side of \eqref{E:2.12} goes to zero as $k\to \infty$, all coefficients of
$q$ must be zero. This immediately implies that $M_{(N-1, 0)} =0 $ and a
simple argument using induction shows that
\begin{equation} \label{E:2.13}
M_{(n, 0)} = 0\; \forall n \;(0 \leq n \leq N-1 ).
\end{equation}
Similarly, replacing $g'(s)ds $ by $dt$ and choosing another sequence of
polynomials
$$( \; p_k(s, t) = \left( \frac{t-b'}{a'-b'}\right)^k \,, \; \;k \geq N
\;)$$
a similar argument gives
\begin{equation}\label{E:2.14}
N_{(0, n)} =0 \; \forall n \;(0\leq n \leq N-1 ).
\end{equation}
Using \eqref{E:2.7}, \eqref{E:2.13} (respectively,
\eqref{E:2.9},
\eqref{E:2.14} ), the assumption $g^\prime \neq 0$ and the induction we
get that $c_i=0
\; \forall i, \;1\leq i
\leq N$ and $d_i=0
\;\forall i\,, \;1\leq i \leq N$. Hence, $u(P)=c_0u(Q)$. The required
result now follows from Theorem
\ref{P:0.1}.
\end{proof}
According to the theorem by L. Schwartz \cite[page 165]{rudin},
if $T $ is a
distribution supported at the point $Q$, then we can find unique constants
$N$, $c_\alpha $ ( $| \alpha | \leq N)$ such that
$$
T = \sum_{| \alpha | \leq N} c_\alpha D^\alpha \delta _Q
\,\,,
$$
where $\delta _Q $ denotes the point evaluation at $Q$.
Using this and Theorem \ref{T:2.1}, we obtain the following partial
negative answer to the point to compact set reflection
conjecture.
\begin{corollary}\label{C:8.1}
Let $P\neq Q$ be points in $\mathbb{R}^2 $ that are sufficiently
close to $\gamma $, where $\gamma $ is a non-singular real analytic curve
which is not a straight line. Then, there is no distribution $T $
supported
at $Q$ such that $u(P) = \langle T , u \rangle \, \forall u,
u_{| _\gamma }=
0 $ and satisfying \eqref{E:0.2}.
\end{corollary}
By using Study's change of variables $z= X+iY,\; w=X-iY$ that reduces the
``real'' Helmholtz's operator $\triangle +\lambda^2 $ to the complex
hyperbolic
operator $4\frac{\partial ^2}{\partial z\partial w} + \lambda^2$,
and applying similar arguments we obtain
\begin{corollary}\label{C:9.3}
Let $P \neq Q $ be points in $\mathbb{R}^2 $that are sufficiently close to
$\gamma $ (but off $\gamma $), $\gamma $ being a non-singular real
analytic curve in $\mathbb{R}^2 $.
(i) If
$$
u(P) = \sum_{| \alpha | \leq N }c_\alpha \, D^\alpha
u (Q)
$$
for all $u$ vanishing on $\gamma $ and satisfying the real Helmholtz's
equation
\begin{equation}\label{E:9.3}
\triangle u + \lambda^2u =0\,,
\end{equation}
where $c_{\alpha } = c_{\alpha }(P, Q)$ and $N=N(P, Q)$ is an
integer, then $c_\alpha = 0$
$\forall \alpha \neq 0 $, $c_0 = -1 $, $P$ and $Q$ must be symmetric with
respect to $\gamma $ and $\gamma $ must be a straight
line.
(ii) If $\gamma $ is not a straight line, then there is no distribution $T$
supported at $Q$ such that $u(P) = \langle T , u \rangle \, \forall u,
u_{| _\gamma }=0 $ and satisfying \eqref{E:9.3}.
\end{corollary}
\begin{remark}\label{R:3.4}
For the operator $L=\frac{\partial^2}{\partial s\partial t}\;$, the
Riemann function is identically equal to 1. Hence, equations
\eqref{E:2.4}-\eqref{E:2.9} are still valid for this case,
with $R \equiv 1$ throughout. Hence, arguing as in the proof of Theorem
\ref{T:2.1}, we obtain that $(i)$ If
$$
u(P) = \sum_{| \alpha | \leq N }c_\alpha \, D^\alpha
u (Q)
$$
for all solutions of $Lu=0$ vanishing on $\gamma $, where $P$ and $Q$ are
two points in $\mathbb{R}^2$ that are sufficiently close to $\gamma $,
$c_\alpha $ = $c_\alpha (P, Q) $ and $N=N(P,Q)$ is an integer, then
$c_\alpha =0$ $\forall $ $\alpha $ $\neq 0 $, $c_0 = -1 $ and $P$ and $Q$
must be symmetric with respect to $\gamma $.
(ii) If
\begin{equation}\label{E:666}
\sum_{| \alpha | \leq N }c_\alpha \, D^\alpha u (Q) = 0
\end{equation}
for all solutions of $Lu=0$ vanishing on $\gamma $, where no
mixed derivatives are involved in \eqref{E:666}, $c_\alpha =
c_\alpha (Q) $ and $N= N(Q)$ is an integer, then $c_\alpha = 0 $ $
\forall \alpha$, $| \alpha | \leq N$.
Similar conclusions hold for $L=\triangle$.
\end{remark}
As it was mentioned in the introduction of this paper, similar to the case
of the two dimensional (real) Helmholtz operator (and when $\gamma $ is
not a straight line), there is no point to point reflection for harmonic
functions in $\mathbb{R}^3 $ with the Schwarz reflection law, unless the
hypersurface under consideration is a hyperplane or a sphere. When the
hypersurface is a cylinder, applying our results and using the ideas in
the proof of Corollary~3.3 in
\cite{khavinsons}, we obtain the following, which in
particular includes that corollary as a special case:
\begin{corollary}\label{C:8.3}
Let $P\neq Q$ be points in $\mathbb{R}^3$ that are sufficiently close to
$\Gamma $ (but off $\Gamma $), where
$$
\Gamma = \{(x_1, x_2, x_3 ) | (x_1, x_2, 0 )\in \gamma ,
\gamma \text{ being a non-singular real analytic curve} \}
$$
is a cylinder in $\mathbb{R}^3 $ with base $\gamma $.
(i) If
\begin{equation}\label{E:3.17}
u(P) = \sum_{| \alpha | \leq N} c_\alpha D^\alpha u(Q)
\end{equation}
for all functions $u$ harmonic near $\Gamma $ and vanishing on $\Gamma $,
where
$c_\alpha = c_\alpha (P,Q)$ and $N=N(P,Q)$ is an integer, then $P$ and $Q$
must be symmetric with respect to $\Gamma $ and $\Gamma $ must be a plane.
(ii) If $\Gamma $ is not a plane, then there is no distribution
$T $ supported at $Q$ such that
$$
u(P) = \langle T , u \rangle
$$
for all $u$ harmonic near $\Gamma $ and vanishing on $\Gamma $.
\end{corollary}
\begin{proof}
(i) Let $P=(x_{1}^{P}, x_{2}^{P}, x_{3}^{P})$, $Q= (x_{1}^{Q},
x_{2}^{Q},x_{3}^{Q})$, $P^0 =(x_{1}^{P}, x_{2}^P)$, $Q^0 = (x_{1}^Q,
x_{2}^Q)$.
Let $u(x_1, x_2, x_3) = v(x_1, x_2), $ where $v$ is harmonic near $\gamma
$ and vanishing on $\gamma $.
Then, $u$ is harmonic near $\Gamma $ and vanishes on $\Gamma $.
Using \eqref{E:3.17}, we obtain,
$$
v(P^0) = \sum_{| \alpha | \leq N} c_\alpha D^\alpha v(Q^0),
$$
for all $v$ harmonic near $\gamma $ and vanishing on $\gamma $.
By Remark $\ref{R:3.4}$, we obtain that
$c_\alpha $ $=$ $0$ $\forall \alpha $ $\neq $ $0$, $c_0 =-1$ and $P^0$ and
$Q^0$ must be symmetric with respect to $\gamma $.
Hence, \eqref {E:3.17}
reduces to
\begin{equation}\label{E:3.18}
u(P) = -u(Q) + \sum_{| \alpha | \leq N}c_\alpha (D^\alpha
u_{x_3})(Q)\,,
\end{equation}
for all $u$ harmonic near $\Gamma $ and vanishing on $\Gamma $.
(Note that from now on, $N$ and the indices of the
constants $c_\alpha $ are in general different from those in
\eqref{E:3.17}).
Next, let $u(x_1, x_2, x_3)$ $ = (x_3 -x_{3}^Q )v(x_1, x_2)$, where $v$
is harmonic near $\gamma $ and vanishing on $\gamma $. Then, $u$ is
harmonic near $\Gamma $ and vanishes on $\Gamma $. Using
\eqref{E:3.18}, we obtain that
$$
(x_{3}^P -x_{3}^Q ) v(P^0) = \sum_{| \alpha | \leq N} c_\alpha
D^\alpha v(Q_0),
$$
for all $v$ harmonic near $\gamma $ and vanishing on $\gamma $.
By Remark \ref{R:3.4}, we find that
$$
c_\alpha =0 \;\forall \alpha \neq 0, \; c_0 = -(x_{3}^P-x_{3}^Q).
$$
Thus, we can write \eqref{E:3.18} (hence, \eqref{E:3.17} ) as
\begin{equation}\label{E:3.19}
u(P) = -u(Q) -(x_{3}^P -x_{3}^Q)u_{x_3}(Q),
\end{equation}
for all $u$ harmonic near $\Gamma $ and vanishing on $\Gamma $.
Finally, let
$$
u(x_1, x_2, x_3) = v(x_1, x_2)e^{\lambda x_3},
$$
where,
\begin{equation}\label{E:3.20}
\triangle_{(x_1, x_2)}v + \lambda^2 v =0, \; v_{|_\gamma }=0 \text{ and
}\lambda >0\,.
\end{equation}
Then, $u$ is harmonic near $\Gamma $ and vanishes on $\Gamma $.
Using \eqref{E:3.19}, we find that
$$
v(P^0) = -e^{-\lambda (x_{3}^P
-x_{3}^Q)}(1+\lambda(x_{3}^P-x_{3}^Q))v(Q^0)\,,
$$
for all $v$ satisfying \eqref{E:3.20}.
Applying Corollary \ref{C:9.3} (or Theorem \ref{P:0.1} ), we obtain that
\begin{equation}\label{E:3.21}
(1+\lambda(x_{3}^P-x_{3}^Q))e^{-\lambda(x_{3}^P -x_{3}^Q) }=1\; \forall
\lambda\; >0\,,
\end{equation}
and that $\gamma $ must be a straight line. Hence,
$\Gamma $ must be a plane.
Moreover, from \eqref{E:3.21}, we must have $x_{3}^P $ = $x_{3}^Q $.
Hence, since we can assume $x_{3}^P = x_{3}^Q =0, $ we conclude that $P$
and $Q$ must be symmetric with respect to $\Gamma $.
(ii) Follows from (i) and the theorem of L. Schwartz (see the remark
preceding Corollary \ref{C:8.1} ).
\end{proof}
\textbf{Acknowledgements. }I would like to thank Professor Dmitry
Khavinson for introducing me to the problem as well as his help in
preparation of this paper.
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\end{document}