On the shape of solutions to elliptic equations in possibly non convex planar domains

TL;DR

This paper investigates the uniqueness of critical points in positive solutions to elliptic equations in planar domains, establishing geometric conditions for uniqueness and analyzing stable solutions in perturbed convex domains.

Contribution

It introduces a geometric condition involving boundary curvature and boundary derivatives to ensure critical point uniqueness and studies stable solutions in nearly convex domains.

Findings

Uniqueness of critical points under specific geometric conditions

Conditions involving boundary curvature and derivatives ensure solution properties

Analysis of stable solutions in perturbed convex domains

Abstract

In this note we prove uniqueness of the critical point for positive solutions of elliptic problems in bounded planar domains: we first examine the Poisson problem - Delta u = f(x,y) finding a geometric condition involving the curvature of the boundary and the normal derivative of f on the boundary to ensure uniqueness of the critical point. In the second part we consider stable solutions of the nonlinear problem -Delta u = f(u) in perturbation of convex domains.