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arXiv · 2309.09283

Regularity for one-phase Bernoulli problems with discontinuous weights and applications

Lorenzo Ferreri·Bozhidar Velichkov
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Abstract

We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension $N\ge 2$, we show the $C^{1, \alpha}$ regularity of the free boundary outside of a singular set of Hausdorff dimension at most $N-3$. In particular, we prove that the free boundaries are $C^{1, \alpha}$ regular in dimension $N=2$, while in dimension $N=3$ the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension $N=2$, which are minimizing for one-phase functionals with weight functions in $L^\infty$ that are arbitrarily close to a positive constant.

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Lorenzo Ferreri, Bozhidar Velichkov. 2023-09-17. Regularity for one-phase Bernoulli problems with discontinuous weights and applications. https://arxiv.org/abs/2309.09283

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