Geodesic
Full and partial regularity for a class of nonlinear free boundary problems
Karakhanyan, Aram1
1 School of Mathematics, The University of Edinburgh, Peter Tait Guthrie Road, EH9 3FD Edinburgh, UK
Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 981-999

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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 χD is the characteristic function of a set D⊂Rn. We compute the second variation of the energy and study the properties of the stability operator. The free boundary ∂{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.

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2020.09.008
Classification : 35R35
Keywords: Free boundary, Regularity theory, Monotonicity formula, Free boundary conditions

Karakhanyan, Aram 1

1 School of Mathematics, The University of Edinburgh, Peter Tait Guthrie Road, EH9 3FD Edinburgh, UK 
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     title = {Full and partial regularity for a class of nonlinear free boundary problems},
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Karakhanyan, Aram. Full and partial regularity for a class of nonlinear free boundary problems. Annales de l'I.H.P. Analyse non linéaire, juillet – août 2021, Tome 38 (2021) no. 4, pp. 981-999. doi: 10.1016/j.anihpc.2020.09.008

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