Regularity for degenerate two-phase free boundary problems
Annales de l'I.H.P. Analyse non linéaire, Tome 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, ${𝒥}_{\gamma }\to \mathrm{min}$, 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 ${𝒥}_{\gamma }$ becomes singular along the free interface $\left\{u=0\right\}$. The degree of singularity is, in turn, dimmed by the parameter $\gamma \in \left[0,1\right]$. 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.

DOI : https://doi.org/10.1016/j.anihpc.2014.03.004
Classification : 35R35,  35J70,  35J75,  35J20
Mots clés : Free boundary problems, Degenerate elliptic operators, Regularity theory
@article{AIHPC_2015__32_4_741_0,
author = {Leit\~ao, Raimundo and de Queiroz, Olivaine S. and Teixeira, Eduardo V.},
title = {Regularity for degenerate two-phase free boundary problems},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
pages = {741--762},
publisher = {Elsevier},
volume = {32},
number = {4},
year = {2015},
doi = {10.1016/j.anihpc.2014.03.004},
zbl = {06476998},
mrnumber = {3390082},
language = {en},
url = {www.numdam.org/item/AIHPC_2015__32_4_741_0/}
}
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, Tome 32 (2015) no. 4, pp. 741-762. doi : 10.1016/j.anihpc.2014.03.004. http://www.numdam.org/item/AIHPC_2015__32_4_741_0/

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