Symmetry and Approximate Symmetry of a Nonlinear Elliptic Problem over a Ring

TL;DR

This paper proves the radial symmetry of solutions to a nonlinear elliptic problem on a perfect ring and demonstrates the stability of approximate symmetry when the domain slightly shifts, using a novel radial correction and evolutionary methods.

Contribution

It introduces a new radial correction technique into the moving plane method and establishes stability results for near-ring domains in nonlinear elliptic problems.

Findings

Radial symmetry of solutions on perfect rings.

Approximate symmetry and stability for slightly shifted domains.

Use of evolutionary perspective to overcome elliptic comparison limitations.

Abstract

A singularly perturbed free boundary problem arising from a real problem associated with a Radiographic Integrated Test Stand concerns a solution of the equation $\Delta u = f(u)$ in a domain $\Omega$ subject to constant boundary data, where the function $f$ in general is not monotone. When the domain $\Omega$ is a perfect ring, we incorporate a new idea of radial correction into the classical moving plane method to prove the radial symmetry of a solution. When the domain is slightly shifted from a ring, we establish the stability of the solution by showing the approximate radial symmetry of the free boundary and the solution. For this purpose, we complete the proof via an evolutionary point of view, as an elliptic comparison principle is false, nevertheless a parabolic one holds.