IUMJ

Title: Non-convexity of level sets in convex rings for semilinear elliptic problems

Authors: Regis Monneau and Henrik Shahgholian

Issue: Volume 54 (2005), Issue 2, 465-472

Abstract:

We show that there is a convex ring $R = \Omega^{-} \setminus \Omega^{+} \subset \mathbb{R}^2$ in which there exists a solution $u$ to a semilinear partial differential equation $\Delta u = f(u)$, $u = -1$ on $\partial\Omega^{-}$, $u = 1$ on $\partial\Omega^{+}$, with level sets, not all convex. Moreover, every bounded solution $u$ has at least one non-convex level set. In our construction, the nonlinearity $f$ is non-positive and smooth.