IUMJ

Title: Quasiconcave solutions to elliptic problems in convex rings

Authors: Chiara Bianchini, Marco Longinetti and Paolo Salani

Issue: Volume 58 (2009), Issue 4, 1565-1590

Abstract:

We investigate the convexity of level sets of solutions to general elliptic equations in a convex ring $\Omega$. In particular, if $u$ is a classical solution which has constant (distinct) values on the two connected components of $\partial\Omega$, we consider its quasi-concave envelope $u^{*}$ (i.e., the function whose superlevel sets are the convex envelopes of those of $u$) and we find suitable assumptions which force $u^{*}$ to be a subsolution of the equation. If a comparison principle holds, this yields $u=u^{*}$ and then $u$ is quasi-concave.