Abstract
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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1541-1601 |
| Number of pages | 61 |
| Journal | Duke Mathematical Journal |
| Volume | 169 |
| Issue number | 8 |
| DOIs | |
| State | Published - 2020 |
Bibliographical note
Funding 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).”
Publisher Copyright:
© 2020 Duke University Press. All rights reserved.