Convexity of level sets for elliptic problems in convex domains or convex rings: two counterexamples (Q2804444)

The very interesting paper under review deals with some geometrical properties of real-valued solutions to Dirichlet problems for semilinear elliptic equations of the form NEWLINENEWLINE\[NEWLINE \Delta u+f(u)=0 NEWLINE\]NEWLINE NEWLINEin a bounded convex domains or convex rings in \(\mathbb{R}^N\) with \(N\geq2.\) 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.NEWLINENEWLINEThe authors prove that the superlevel sets of the solutions do not always inherit the convexity or ring-convexity of the domain. Namely, two counterexamples are given to this quasiconcavity property: the first one for some two-dimensional convex domains and the second one for some convex rings in any dimension.