Generic properties in free boundary problems
X Fernández-Real, H Yu - arXiv preprint arXiv:2308.13209, 2023 - arxiv.org
X Fernández-Real, H Yu
arXiv preprint arXiv:2308.13209, 2023•arxiv.orgIn 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 …
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 …
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 . More precisely, we show that for a co-countable subset of , minimizers have smooth free boundaries in for the Alt-Caffarelli and in 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 .
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