## Abstract

In this paper we study the free boundary regularity for almost-minimizers of the functional J(u)=∫Ω|∇u(x)| ^{2} +q _{+} ^{2} (x)χ _{{u>0}} (x)+q _{−} ^{2} (x)χ _{{u<0}} (x)dx where q _{±} ∈L ^{∞} (Ω). Almost-minimizers satisfy a variational inequality but not a PDE or a monotonicity formula the way minimizers do (see [4], [5], [9], [37]). Nevertheless, using a novel argument which brings together tools from potential theory and geometric measure theory, we succeed in proving that, under a non-degeneracy assumption on q _{±} , the free boundary is uniformly rectifiable. Furthermore, when q _{−} ≡0, and q _{+} is Hölder continuous we show that the free boundary is almost-everywhere given as the graph of a C ^{1,α} function (thus extending the results of [4] to almost-minimizers).

Original language | English (US) |
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Pages (from-to) | 1109-1192 |

Number of pages | 84 |

Journal | Advances in Mathematics |

Volume | 350 |

DOIs | |

State | Published - Jul 9 2019 |

### Bibliographical note

Funding Information:G. David was partially supported by the Institut Universitaire de France, the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014 and the Simons Collaborations in MPS Grant 601941 GD. M. Engelstein was partially supported by an NSF Graduate Research Fellowship, NSF DGE 1144082, the University of Chicago RTG grant DMS 1246999, a NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison's grant NSF DMS 1500771. T. Toro was partially supported by a Guggenheim fellowship, NSF grant DMS-1361823, by the Robert R. & Elaine F. Phelps Professorship in Mathematics and by the Craig McKibben & Sarah Merner Professorship in Mathematics. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.☆ G. David was partially supported by the Institut Universitaire de France, the ANR, programme blanc GEOMETRYA, ANR-12-BS01-0014 and the Simons Collaborations in MPS Grant 601941 GD. M. Engelstein was partially supported by an NSF Graduate Research Fellowship, NSF DGE 1144082, the University of Chicago RTG grant DMS 1246999, a NSF postdoctoral fellowship, NSF DMS 1703306 and by David Jerison's grant NSF DMS 1500771. T. Toro was partially supported by a Guggenheim fellowship, NSF grant DMS-1361823, by the Robert R. & Elaine F. Phelps Professorship in Mathematics and by the Craig McKibben & Sarah Merner Professorship in Mathematics. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.

Publisher Copyright:

© 2019

## Keywords

- Almost-minimizer
- Free boundary problem
- Uniform rectifiability