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

2023/01/19 by Luca Battaglia, Battaglia, Luca, Fabio De Regibus +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2301.08098

openalex publication_date 2023/01/19 · openalex created_date 2023/01/21 · openalex updated_date 2026/07/28

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.

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