JournalsaihpcVol. 38, No. 4pp. 981–999

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

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

J_{F}(u) = \int \limits_{\mathrm{\Omega }}F(|\mathrm{∇}u|\right.^{2} + \lambda ^{2}\chi _{\{u > 0\}},\right.\right.

where F satisfies some structural conditions and χD\chi _{D} is the characteristic function of a set DRnD \subset \mathbb{R}^{n}. We compute the second variation of the energy and study the properties of the stability operator. The free boundary {u>0}\partial \{u > 0\} can be seen as a rectifiable n1n−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=3n = 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