Interface regularity for semilinear one-phase problems

Alessandro Audrito; Joaquim Serra

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arXiv · 2110.09210

Interface regularity for semilinear one-phase problems

Alessandro Audrito·Joaquim Serra

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Abstract

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Bernoulli's free boundary problem, also known as one-phase problem. We prove a $C^{1,\alpha}$ estimates for the "interfaces" (level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole $\mathbb{R}^N$, for $N \leq 4$, answering positively a conjecture of Fern\'andez-Real and Ros-Oton. Our results are to Bernoulli's free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces.

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Alessandro Audrito, Joaquim Serra. 2021-10-18. Interface regularity for semilinear one-phase problems. https://arxiv.org/abs/2110.09210

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