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arXiv · 1508.07447

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

Serena Dipierro·Aram L. Karakhanyan
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Abstract

In this paper we study the two-phase Bernoulli type free boundary problem arising from the minimization of the functional $$ J(u):=\int_Ω|\nabla u|^p +λ_+^p\,χ_{\{u>0\}} +λ_-^p\,χ_{\{u\le 0\}}, \quad 1 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 $p=2$.

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Serena Dipierro, Aram L. Karakhanyan. 2015-12-10. Stratification of free boundary points for a two-phase variational problem. https://arxiv.org/abs/1508.07447

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