On symmetry problems \thanks{key words: symmetry, Pompeiu problem, Schiffer's conjecture, Helmholtz equations } \thanks{AMS subject classification: 35R30, 35J40, 35J05} %PACS 02.30.Jr, 03.40.Kf
A.G. Ramm
Mathematics Department
Manhattan
KS 66506-2602
USA
Tis paper is withdrawn. Several symmetry problems are discussed.
math0504035.tex
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\begin{document}
%A.G.Ramm,  
%\begin{titlepage}

\title{ On symmetry problems
   \thanks{key words:  symmetry, Pompeiu problem, Schiffer's conjecture, 
Helmholtz equations
    }
   \thanks{AMS subject classification: 35R30, 35J40, 35J05} 
%PACS 02.30.Jr, 03.40.Kf   }
}

\author{
A.G. Ramm\\
Mathematics Department, 
Kansas State University, \\
 Manhattan, KS 66506-2602, USA\\
ramm@math.ksu.edu\\
}

\date{}

\maketitle\thispagestyle{empty}

\begin{abstract} 
Tis paper is withdrawn.
Several symmetry problems are discussed. 
\end{abstract}


%\end{titlepage}


\section{Introduction}
There is a large literature on symmetry problems for PDE, in particular, 
for the Helmholtz equation. The most well-known and still open problems
are the Pompeiu problem and Schiffer's conjecture, which are stated below.
In this paper some old and new conjectures are formulated. One of them, 
Conjecture A, is proved under assumption that Conjecture P holds.
 It is proved that if a solution $u$ to 
Helmholtz equation in a domain $D$, homeomorphic to a ball, is spherically 
symmetric in some ball $B(x_0,a)\subset D$, centered at a point $x_0\in 
D$ of radius $a>0$, and $u|_{\partial D}=const$, then $D$ is a ball
(Lemma 1). An analog of Conjecture P is proved (Lemma 2). A symmetry 
question related to a wave equation, is discussed.



\end{document}



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