Regularity for degenerate two-phase free boundary problems
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, pp. 741-762.

We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase free boundary problems, 𝒥 γ min , ruled by nonlinear, p-degenerate elliptic operators. Included in such family are heterogeneous cavitation problems of Prandtl–Batchelor type, singular degenerate elliptic equations; and obstacle type systems. The Euler–Lagrange equation associated to 𝒥 γ becomes singular along the free interface {u=0}. The degree of singularity is, in turn, dimmed by the parameter γ[0,1]. For 0<γ<1 we show that local minima are locally of class C 1,α for a sharp α that depends on dimension, p and γ. For γ=0 we obtain a quantitative, asymptotically optimal result, which assures that local minima are Log-Lipschitz continuous. The results proven in this article are new even in the classical context of linear, nondegenerate equations.

DOI: 10.1016/j.anihpc.2014.03.004
Classification: 35R35,  35J70,  35J75,  35J20
Keywords: Free boundary problems, Degenerate elliptic operators, Regularity theory
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Leitão, Raimundo; de Queiroz, Olivaine S.; Teixeira, Eduardo V. Regularity for degenerate two-phase free boundary problems. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 4, pp. 741-762. doi : 10.1016/j.anihpc.2014.03.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.03.004/

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