Convexity of level sets for elliptic problems in convex domains or convex rings: two counterexamples

TL;DR

This paper presents two counterexamples showing that solutions to certain elliptic equations in convex domains or rings do not necessarily have convex superlevel sets, challenging assumptions about geometric properties of solutions.

Contribution

It provides the first known counterexamples demonstrating that quasiconcavity of solutions is not guaranteed in convex or convex ring domains.

Findings

Counterexamples in 2D convex domains

Counterexamples in convex rings of any dimension

Superlevel sets may lack convexity despite domain convexity

Abstract

This paper deals with some geometrical properties of solutions of some semilinear elliptic equations in bounded convex domains or convex rings. Constant boundary conditions are imposed on the single component of the boundary when the domain is convex, or on each of the two components of the boundary when the domain is a convex ring. A function is called quasiconcave if its superlevel sets, defined in a suitable way when the domain is a convex ring, are all convex. In this paper, we prove that the superlevel sets of the solutions do not always inherit the convexity or ring-convexity of the domain. Namely, we give two counterexamples to this quasiconcavity property: the first one for some two-dimensional convex domains and the second one for some convex rings in any dimension.