Regularity for degenerate two-phase free boundary problems

Leitão, Raimundo ; de Queiroz, Olivaine S. ; Teixeira, Eduardo V.

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015), p. 741-762 / Harvested from Numdam

Résumé

We provide a rather complete description of the sharp regularity theory to a family of heterogeneous, two-phase free boundary problems, $ {\mathcal{J}}_{\gamma }\to \mathrm{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 $ {\mathcal{J}}_{\gamma }$ becomes singular along the free interface $ \{u=0\}$. The degree of singularity is, in turn, dimmed by the parameter $ \gamma \in [0,1]$. For $ 0<\gamma <1$ we show that local minima are locally of class $ {C}^{1,\alpha }$ for a sharp α that depends on dimension, p and γ. For $ \gamma =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.

Détail

Publié le : 2015-01-01

MR 3390082 | Zbl 06476998

DOI : https://doi.org/10.1016/j.anihpc.2014.03.004
Classification:  35R35,  35J70,  35J75,  35J20

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