A Bernoulli problem with non-constant gradient boundary constraint

Abstract

In this article, we present a result about the existence and convexity of solutions to a free boundary problem of Bernoulli type, with non-constant gradient boundary constraint depending on the outer unit normal. In particular, we prove that, in the convex case, the existence of a subsolution guarantees the existence of a classical solution, which is proved to be convex.

Keywords:

• Bernoulli problem
• convexity

AMS Subject Classifications::

• 35R35
• 35J66
• 35J70

Acknowledgements

The author wishes to thank Paolo Salani warmly for his invaluable help, several helpful discussions and, above all, for his encouragement and support.