### 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 |

Externally published | Yes |

### Keywords

- Almost-minimizer
- Free boundary problem
- Uniform rectifiability

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## Cite this

*Advances in Mathematics*,

*350*, 1109-1192. https://doi.org/10.1016/j.aim.2019.04.059