A short survey on the convexity of level sets for divergence-form elliptic equations

Abstract

Let $Ω\subset\mathbb{R}^n$ be a bounded, smooth and (uniformly) convex domain, and let $u$ be a positive solution to an elliptic boundary value problem with zero Dirichlet data. A classical problem in the analysis of partial differential equations is to determine whether the convexity of $Ω$ is inherited by the superlevel (or sublevel) sets of $u$. This survey considers divergence-form equations in bounded convex domains. For torsion functions, first eigenfunctions, and semilinear equations with sublinear nonlinearities, concavity of suitable transforms of $u$ gives positive results. Two further methods, i.e. envelope constructions and constant-rank theorems for level-set curvature, often provide alternative routes to convexity and related properties. These results are complemented by counterexamples, showing that in general convexity of the domain and ellipticity of the operator do not guarantee quasiconvexity for every solution. We present these counterexamples through a shared underlying idea, and describe the distinct ways in which different equations implement this idea.