Positive solutions to general semilinear overdetermined boundary problems

TL;DR

This paper proves the existence of positive solutions for a broad class of overdetermined semilinear elliptic boundary problems on bounded domains in low dimensions, extending previous variational methods and addressing challenges related to uniform bounds and free boundary regularity.

Contribution

It introduces a new approach to establish positive solutions for overdetermined elliptic problems in dimensions up to four, handling variable coefficients and nonlinearities with mild assumptions.

Findings

Existence of solutions on domains of any prescribed volume in low dimensions.

Extension of methods to higher dimensions with potential singular free boundaries.

Results are novel even for classical Poisson equations with Neumann boundary conditions.

Abstract

We establish the existence of positive solutions to a general class of overdetermined semilinear elliptic boundary problems on suitable bounded open sets $\Omega\subset\mathbb{R}^n$. Specifically, for $n\leq 4$ and under mild technical hypotheses on the coefficients and the nonlinearity, we show that there exist open sets $\Omega\subset\mathbb{R}^n$ with smooth boundary and of any prescribed volume where the overdetermined problem admits a positive solution. The proof builds on ideas of Alt and Caffarelli on variational problems for functions defined on a bounded region. In our case, we need to consider functions defined on the whole $\mathbb{R}^n$, so the key challenge is to obtain uniform bounds for the minimizer and for the diameter of its support. Our methods extend to higher dimensions, although in this case the free boundary $\partial\Omega$ could have a singular set of codimension 5. The results are new even in the case of the Poisson equation $-\Delta v =g(x)$ with constant Neumann data.