Overdetermined boundary problems with nonconstant Dirichlet and Neumann data

TL;DR

This paper proves the existence of nontrivial solutions to overdetermined boundary problems for second order semilinear elliptic equations with nonconstant boundary data and position-dependent coefficients, expanding classical symmetry results.

Contribution

It demonstrates that solutions exist even when boundary data and coefficients vary spatially, generalizing previous results that required constant data.

Findings

Nontrivial solutions exist under minor technical conditions.

Solutions can be constructed without radial symmetry.

The work extends overdetermined problem theory to variable data and coefficients.

Abstract

In this paper we consider the overdetermined boundary problem for a general second order semilinear elliptic equation on bounded domains of $\mathbf{R}^n$, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.