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In this paper we classify the nonnegative global minimizers of the functional
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.
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. 981–999DOI 10.1016/J.ANIHPC.2020.09.008