## Abstract / Description of output

In this paper we classify the nonnegative global minimizers of the functional

JF(u)=∫ΩF(|∇u|2)+λ2χ{u>0},

where F satisfies some structural conditions and is the characteristic function of a set. We compute the second variation of the energy and study the properties of the stability operator. The free boundary can be seen as a rectifiable varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.

JF(u)=∫ΩF(|∇u|2)+λ2χ{u>0},

where F satisfies some structural conditions and is the characteristic function of a set. We compute the second variation of the energy and study the properties of the stability operator. The free boundary can be seen as a rectifiable varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded. Hence one can use Allard's monotonicity formula to prove the existence of tangent cones modulo a set of small Hausdorff dimension. In particular, we prove that if and the ellipticity constants of the quasilinear elliptic operator generated by F are close to 1 then the conical free boundary must be flat.

Original language | English |
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Pages (from-to) | 981-999 |

Journal | Annales de l'Institut Henri Poincaré C |

Volume | 38 |

Issue number | 4 |

Early online date | 5 Oct 2020 |

DOIs | |

Publication status | Published - 31 Aug 2021 |