On the uniqueness and monotonicity of solutions of free boundary problems

TL;DR

This paper proves the uniqueness and monotonicity of solutions to free boundary problems in bounded domains, extending results beyond two-dimensional balls and solving a long-standing open problem in the field.

Contribution

It establishes the first known uniqueness results for free boundary problems in non-ball domains for p>1 and introduces universal monotonicity properties of solutions.

Findings

Uniqueness of positive solutions in smooth bounded domains depending on Sobolev constants.

Monotonic behavior of boundary density and energy sharing universal properties.

Complete description of solution branches on a 2D ball, including monotonicity until boundary density vanishes.

Abstract

For any $\Omega\subset \mathbb{R}^N$ smooth and bounded domain, we prove uniqueness of positive solutions of free boundary problems arising in plasma physics on $\Omega$ in a neat interval depending only by the best constant of the Sobolev embedding $H^{1}_0(\Omega)\hookrightarrow L^{2p}(\Omega)$, $p\in [1,\frac{N}{N-2})$ and show that the boundary density and a suitably defined energy share a universal monotonic behavior. At least to our knowledge, for $p>1$, this is the first result about the uniqueness for a domain which is not a two-dimensional ball and in particular the very first result about the monotonicity of solutions, which seems to be new even for $p=1$. The threshold, which is sharp for $p=1$, yields a new condition which guarantees that there is no free boundary inside $\Omega$. As a corollary, in the same range, we solve a long-standing open problem (dating back to the work of Berestycki-Brezis in 1980) about the uniqueness of variational solutions. Moreover, on a two-dimensional ball we describe the full branch of positive solutions, that is, we prove the monotonicity along the curve of positive solutions until the boundary density vanishes.