We prove the uniqueness of blowups and C 1;log-regularity for the free-boundary of minimizers of the Alt–Caffarelli functional at points where one blowup has an isolated singularity. We do this by establishing a (log-)epiperimetric inequality for the Weiss energy for traces close to that of a cone with isolated singularity, whose free boundary is graphical and smooth over that of the cone in the sphere. With additional assumptions on the cone, we can prove a classical epiperimetric inequality which can be applied to deduce a C 1;α-regularity result. We also show that these additional assumptions are satisfied by the De Silva–Jerison-type cones, which are the only known examples of minimizing cones with isolated singularity. Our approach draws a connection between epiperimetric inequalities and the Łojasiewicz inequality, and, to our knowledge, provides the first regularity result at singular points in the one-phase Bernoulli problem.
Bibliographical noteFunding Information:
Engelstein’s work was partially supported by National Science Foundation (NSF) Mathematical Sciences Postdoctoral Research Fellowships (MSPRF) grant DMS-1703306. Spolaor’s work was partially supported by NSF grant DMS-1810645. Velichkov’s work was partially supported by Agence Nationale de la Recherche (ANR) LabEx PERSYVAL-Lab GeoSpec project grant ANR-11-LABX-0025-01 and by ANR project “Convergent Metrics for Digital Calculus (CoMeDiC).”
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