We demonstrate that a method of Colesanti and Salani, which compares solutions of elliptic differential equations to their quasiconcave envelopes, can be extended to derive convexity of free boundaries. As examples we present the so-called dam problem, a free boundary problem modelling pollution and a Bernoulli problem. Moreover, we prove strict convexity of the wet region in the dam problem in arbitrary dimensions.
Cite this article
Bernd Kawohl, Antonio Greco, On the convexity of some free boundaries. Interfaces Free Bound. 11 (2009), no. 4, pp. 503–514