# Full and partial regularity for a class of nonlinear free boundary problems

### Aram Karakhanyan

School of Mathematics, The University of Edinburgh, Peter Tait Guthrie Road, EH9 3FD Edinburgh, UK

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## Abstract

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

where *F* satisfies some structural conditions and $\chi _{D}$ is the characteristic function of a set $D \subset \mathbb{R}^{n}$. We compute the second variation of the energy and study the properties of the stability operator. The free boundary $\partial \{u > 0\}$ can be seen as a rectifiable $n−1$ 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 $n = 3$ and the ellipticity constants of the quasilinear elliptic operator generated by *F* are close to 1 then the conical free boundary must be flat.

## Cite this article

Aram Karakhanyan, Full and partial regularity for a class of nonlinear free boundary problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 4, pp. 981–999

DOI 10.1016/J.ANIHPC.2020.09.008