Non-convexity of level sets in convex rings for semilinear elliptic problems Regis MonneauH. Shahgholian 35J6035R35non-convexitylevel setsemilinear elliptic equationconvex ring 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. Indiana University Mathematics Journal 2005 text pdf 10.1512/iumj.2005.54.2514 10.1512/iumj.2005.54.2514 en Indiana Univ. Math. J. 54 (2005) 465 - 472 state-of-the-art mathematics http://iumj.org/access/