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

2013/04/11 by François Hamel, Nikolaï Nadirashvili, Hamel, Francois +3 · 4 citations
Computer Science · Mathematics · #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · doi:10.48550/arxiv.1304.3355

openalex publication_date 2013/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Least total curvature solutions to steady Euler system and monotone solutions to semilinear equations in a strip
• Log-concavity and anti-maximum principles for semilinear and linear elliptic equations

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