QUANTITATIVE STRATIFICATION FOR SOME FREE-BOUNDARY PROBLEMS

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.