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

Serena Dipierro, Aram L. Karakhanyan

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional J(u):=∫Ω|∇u|p+λp+χ{u>0}+λp−χ{u≤0},1<p<∞. Here $\Omega \subset \R^N$ is a bounded smooth domain and λ± are positive constants such that λp+−λp−>0. We prove the following dichotomy: if x0 is a free boundary point then either the free boundary is smooth near x0 or u has linear growth at x0. 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.
Original languageEnglish
Pages (from-to)40-81
Number of pages33
JournalAdvances in Mathematics
Volume328
Early online date12 Jan 2018
DOIs
Publication statusPublished - 13 Apr 2018

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