Abstract
The results of this paper allow one to derive several results of general interest: to prove the Schiffer’s conjecture, to solve the Pompeiu problem, to prove two symmetry results in harmonic analysis and to give a new method for solving an old symmetry problem.
Highlights
Let D be a bounded connected domain in Rm, m ≥ 2, with a smooth boundary S, N is the outer unit normal to S, k > 0 is a constant, uN := uN|S is the limiting value of the normal derivative of u on S from D
Various symmetry problems were considered by the author in [1,2,3,4,5,6]
One can derive from Theorem 1 the following symmetry results in harmonic analysis
Summary
Let D be a bounded connected domain in Rm, m ≥ 2, with a smooth boundary S, N is the outer unit normal to S, k > 0 is a constant, uN := uN|S is the limiting value of the normal derivative of u on S from D. By different methods, this problem was treated in [2], and in [7].
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