Solution to the Pompeiu problem and the related symmetry problem

Abstract

Assume that D ⊂ R 3 is a bounded domain with C 1 −smooth boundary. Our result is:Four equivalent formulations of the Pompeiu problem are discussed.A domain D has P −property if there exists an f = 0, f ∈ L 1 loc (R 3 ) such that D f (gx + y)dx = 0 for all y ∈ R 3 and all g ∈ SO(2), where SO(2) is the rotation group.The result obtained concerning the related symmetry problem is: Theorem 2. If (∇ 2 + k 2 )u = 0 in D, u| S = 1, u N | S = 0, and k > 0 is a constant, then D is a ball.

On the discrete Fuglede and Pompeiu problems

“…On the other hand, for every nonempty polygon (moreover, for any convex domain with at least one corner) the answer for Question 1 is affirmative by the result of Brown, Taylor and Schreiber [2]. Recently, Ramm [28] showed that there exists a f ≡ 0 function that satisfies the 3-dimensional analogue of (2) for a bounded domain K ⊆ R 3 with C 1 -smooth boundary if and only if K is a closed ball. Extensive literature is concerned with the Pompeiu problem.…”

Functions of constant geodesic x-ray transform

“…This is a conjecture stating that if a non-zero continuous function integrates to zero over every congruent copy of a simply connected Lipschitz domain, then the domain must be a ball. For a more detailed description of the problem and a verification of the conjecture in three dimensions, see [13].…”

Inverse problem of potential theory

“…Lemma 3 is proved. ✷ Lemma 3 is Lemma 11.2.2 in [3], see also Theorem 2 in [8] and [10]. Its short proof is included for convenience of the reader.…”