Nonlocal minimal surfaces are unique for all but countably many parameters in any strictly increasing family of exterior data, and arbitrarily small perturbations yield smooth minimizers in one dimension beyond the critical threshold.
2308.13209 , archivePrefix=
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abstract
In this work, we show the generic uniqueness of minimizers for a large class of energies, including the Alt-Caffarelli and Alt-Phillips functionals. We then prove the generic regularity of free boundaries for minimizers of the one-phase Alt-Caffarelli and Alt-Phillips functionals, for a monotone family of boundary data $\{\varphi_t\}_{t\in(-1,1)}$. More precisely, we show that for a co-countable subset of $\{\varphi_t\}_{t\in(-1,1)}$, minimizers have smooth free boundaries in $\mathbb{R}^5$ for the Alt-Caffarelli and in $\mathbb{R}^3$ for the Alt-Phillips functional. In general dimensions, we show that the singular set is one dimension smaller than expected for almost every boundary datum in $\{\varphi_t\}_{t\in(-1,1)}$.
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Nonlocal minimal surfaces are generically unique, and smooth in one extra dimension
Nonlocal minimal surfaces are unique for all but countably many parameters in any strictly increasing family of exterior data, and arbitrarily small perturbations yield smooth minimizers in one dimension beyond the critical threshold.