Two-phase almost minimizers for a fractional free boundary problem

Mark Allen; Mariana Smit Vega Garcia

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arXiv · 2308.06357

Two-phase almost minimizers for a fractional free boundary problem

Mark Allen·Mariana Smit Vega Garcia

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Abstract

In this paper, we study almost minimizers to a fractional Alt-Caffarelli-Friedman type functional. Our main results concern the optimal $C^{0,s}$ regularity of almost minimizers as well as the structure of the free boundary. We first prove that the two free boundaries $F^+(u)=\partial\{u(\cdot,0)>0\}$ and $F^-(u)=\partial\{u(\cdot,0)<0\}$ cannot touch, that is, $F^+(u)\cap F^-(u)=\emptyset$. Lastly, we prove a flatness implies $C^{1,\gamma}$ result for the free boundary.

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Mark Allen, Mariana Smit Vega Garcia. 2023-08-11. Two-phase almost minimizers for a fractional free boundary problem. https://arxiv.org/abs/2308.06357

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