Symmetry problem

Abstract

A novel approach to an old symmetry problem is developed. A new proof is given for the following symmetry problem, studied earlier: if Δu = 1 in D ⊂ R 3 , u = 0 on S, the boundary of D, and u N = const on S, then S is a sphere. It is assumed that S is a Lipschitz surface homeomorphic to a sphere. This result has been proved in different ways by various authors. Our proof is based on a simple new idea.

Solution to the Pompeiu problem and the related symmetry problem

“…Define w := J j=1 h j u j (x). Then w solves equation (3) and w = w N = 0 on S. By the uniqueness of the solution to the Cauchy problem for equation (3) it follows that w = 0 in D. Since the set {u j (x)} J j=1 is linearly independent in L 2 (D), one gets h j = 0, 1 ≤ j ≤ J. Lemma 2 is proved. Lemma 3.…”

“…If an entire function of exponential typeχ(ξ) vanishes on the irreducible algebraic variety ξ 2 = k 2 , then the functionũ :=χ(ξ)(ξ 2 − k 2 ) −1 is also entire and of the same exponential type. Its Fourier transform u(x) solves problem (3). The function u is defined in all of R 3 , u ∈ H 2 loc (R 3 ) and is compactly supported by the Paley-Wiener theorem.…”

A problem in analysis

“…Using the method from [2] (see also [3]) we derive from (2.4) that D is a disc. It follows from (2.4) that the harmonic in D 0 D R 2 n D function It follows from (2.5) that the functions u.x/ and u N .x/ are constant on C R , since the normal N on C R is directed along the radius.…”

“…N 1 s 2 C N 2 s 1 /hds D 0; (2.11) where N j , j D 1; 2, are the components of the outer unit normal N to C . It is proved in [2] that the set of restrictions of all harmonic functions in B R , regular at the origin, onto a closed curve C B R , diffeomorphic to a circle, is dense in L 2 .C /. Therefore, (2.11) implies N 1 s 2 C N 2 s 1 D 0 8s 2 C: (2.12)…”

Inverse problem of potential theory

“…In [3], p.334, a new proof of Proposition 1 was given. In [4]- [9], see also [11], some inverse problems and symmetry problems are studied.…”