no. 4
The two membranes problem for different operators
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 899-932.

We study the two membranes problem for different operators, possibly nonlocal. We prove a general result about the Hölder continuity of the solutions and we develop a viscosity solution approach to this problem. Then we obtain C1,γ regularity of the solutions provided that the orders of the two operators are different. In the special case when one operator coincides with the fractional Laplacian, we obtain the optimal regularity and a characterization of the free boundary.

DOI : 10.1016/j.anihpc.2016.05.006
Mots clés : Obstacle problem, Fractional operators, Regularity 
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     title = {The two membranes problem for different operators},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {899--932},
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Caffarelli, L.; De Silva, D.; Savin, O. The two membranes problem for different operators. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 899-932. doi : 10.1016/j.anihpc.2016.05.006. http://www.numdam.org./articles/10.1016/j.anihpc.2016.05.006/

[1] Azevedo, A.; Rodrigues, J.-F.; Santos, L. The N-membranes problem for quasilinear degenerate systems, Interfaces Free Bound., Volume 7 (2005) no. 3, pp. 319–337 | MR | Zbl

[2] Caffarelli, L.A.; Cabre, X. Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995 (English summary) vi+104 pp | DOI | MR | Zbl

[3] Caffarelli, L.A.; Chan, C-H.; Vasseur, A. Regularity theory for parabolic nonlinear integral operators, J. Am. Math. Soc., Volume 24 (2011) no. 3, pp. 849–869 | DOI | MR | Zbl

[4] Caffarelli, L.A.; De Silva, D.; Savin, O. Obstacle type problems for non-local minimal surfaces, Commun. Partial Differ. Equ., Volume 41 (2016) no. 8, pp. 1303–1323 | DOI | MR | Zbl

[5] Caffarelli, L.A.; Salsa, S.; Silvestre, L. Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian, Invent. Math., Volume 171 (2008) no. 2, pp. 425–461 | DOI | MR | Zbl

[6] Caffarelli, L.A.; Silvestre, L. Regularity theory for fully nonlinear integro-differential equations, Commun. Pure Appl. Math., Volume 62 (2009) no. 5, pp. 597–638 | DOI | MR | Zbl

[7] Caffarelli, L.A.; Silvestre, L. Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., Volume 200 (2011), pp. 59–88 | DOI | MR | Zbl

[8] Carillo, S.; Chipot, M.; Vergara-Caffarelli, G. The N-membrane problem with nonlocal constraints, J. Math. Anal. Appl., Volume 308 (2005) no. 1, pp. 129–139 | DOI | MR | Zbl

[9] Chipot, M.; Vergara-Caffarelli, G. The N-membranes problem, Appl. Math. Optim., Volume 13 (1985) no. 3, pp. 231–249 | MR

[10] Guillen, N. Optimal regularity for the Signorini problem, Calc. Var. Partial Differ. Equ., Volume 36 (2009) no. 4, pp. 533–546 | DOI | MR | Zbl

[11] Kriventsov, D. C1,α interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Commun. Partial Differ. Equ., Volume 38 (2013), pp. 2081–2106 | DOI | MR | Zbl

[12] Ros-Oton, X.; Serra, J. Boundary regularity for fully nonlinear integro-differential equations, Duke Math. J., Volume 165 (2016) no. 11, pp. 2079–2154 | DOI | MR | Zbl

[13] Savin, O. Small perturbation solutions for elliptic equations, Commun. Partial Differ. Equ., Volume 32 (2007) no. 4–6, pp. 557–578 | MR | Zbl

[14] Serra, J. Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels | arXiv | DOI | Zbl

[15] Silvestre, L. The two membranes problem, Commun. Partial Differ. Equ., Volume 30 (2005) no. 1–3, pp. 245–257 | MR | Zbl

[16] Silvestre, L. Regularity of the obstacle problem for a fractional power of the Laplace operator, Commun. Pure Appl. Math., Volume 60 (2007) no. 1, pp. 67–112 | DOI | MR | Zbl

[17] Vergara-Caffarelli, G. Regolarita di un problema di disequazioni variazionali relativo a due membrane, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. (8), Volume 50 (1971), pp. 659–662 (Italian, with English summary) | MR | Zbl

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