On symmetry problems

TL;DR

The paper investigates symmetry phenomena for partial differential equations, focusing on classic open problems like the Pompeiu problem and Schiffer conjecture while formulating new conjectures. A key result is that Conjecture A follows from Conjecture P; in addition, Lemma 1 proves that a Helmholtz solution in a ball-like domain with local spherical symmetry and constant boundary data forces the domain to be a ball. An analogue of Conjecture P is established (Lemma 2), and the discussion extends to a symmetry question for the wave equation. By linking geometric domain properties to solution symmetries, the work contributes to inverse-type uniqueness questions in PDE.

Abstract

withdrawn Several symmetry problems are discussed. These include the Pompeiu problem and similar conjectures for the heat and wave equations.