Quantitative stratification for some free-boundary problems

Edelen, Nick; Engelstein, Max

doi:10.1090/tran/7401

Quantitative stratification for some free-boundary problems
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by Nick Edelen and Max Engelstein PDF

Trans. Amer. Math. Soc. 371 (2019), 2043-2072 Request permission

Abstract:

In this paper we prove the rectifiability of and measure bounds on the singular set of the free-boundary for minimizers of a functional first considered by Alt–Caffarelli [J. Reine Angew. Math. 325 (1981), pp. 105–144]. Our main tools are the Quantitative Stratification and Rectifiable-Reifenberg framework of Naber–Valtorta [Ann. of Math. (2) 185 (2017), pp. 131–227], which allow us to do a type of “effective dimension-reduction”. The arguments are sufficiently robust that they apply to a broad class of related free-boundary problems as well.

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