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.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 2043-2072 |
| Number of pages | 30 |
| Journal | Transactions of the American Mathematical Society |
| Volume | 371 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2019 |
| Externally published | Yes |
Bibliographical note
Funding Information:Received by the editors February 24, 2017, and, in revised form, September 12, 2017. 2010 Mathematics Subject Classification. Primary 35R35. The first author was supported by NSF grant DMS-1606492. The second author was partially supported by NSF Grant No. DMS-1440140 while the author was in residence at MSRI in Berkeley, California, during Spring 2017.
Publisher Copyright:
© 2018 American Mathematical Society.