Stratification of free boundary points for a two-phase variational problem

TL;DR

This paper investigates the regularity and structure of free boundary points in a two-phase variational problem involving the p-Laplacian, establishing conditions for smoothness, linear growth, and measure-theoretic properties of the free boundary.

Contribution

The paper introduces a new approach to analyze free boundary regularity for the p-Laplacian case, extending classical results and establishing measure-theoretic properties of the free boundary.

Findings

Free boundary near a point is either smooth or the solution has linear growth.

The free boundary has locally finite perimeter.

The set of non-smooth free boundary points has zero Hausdorff measure.

Abstract

In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional $$ J(u):=\int_{\Omega}|\nabla u|^p +\lambda_+^p\,\chi_{\{u>0\}} +\lambda_-^p\,\chi_{\{u\le 0\}}, \quad 1<p<\infty. $$ Here $\Omega \subset \R^N$ is a bounded smooth domain and $\lambda_\pm$ are positive constants such that $\lambda_+^p-\lambda^p_->0$. We prove the following dichotomy: if $x_0$ is a free boundary point then either the free boundary is smooth near $x_0$ or $u$ has linear growth at $x_0$. Furthermore, we show that for $p>1$ the free boundary has locally finite perimeter and the set of non-smooth points of free boundary is of zero $(N-1)$-dimensional Hausdorff measure. Our approach is new even for the classical case $p=2$.