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arXiv · 2210.03552

Rectifiability and uniqueness of blow-ups for points with positive Alt-Caffarelli-Friedman limit

Mark Allen·Dennis Kriventsov·Robin Neumayer
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Abstract

We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt-Caffarelli-Friedman (ACF) monotonicity formula is asymptotically positive forms an $\mathcal{H}^{n-1}$-rectifiable set. Moreover, for $\mathcal{H}^{n-1}$-a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in $W^{1,2}$ to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber-Valtorta framework and our recent result establishing a sharp quantitative remainder term in the ACF monotonicity formula. We also give applications of our results to free boundary problems.

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Mark Allen, Dennis Kriventsov, Robin Neumayer. 2022-10-07. Rectifiability and uniqueness of blow-ups for points with positive Alt-Caffarelli-Friedman limit. https://arxiv.org/abs/2210.03552

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