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 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.
@article{AIHPC_2017__34_4_899_0, author = {Caffarelli, L. and De Silva, D. and Savin, O.}, title = {The two membranes problem for different operators}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {899--932}, publisher = {Elsevier}, volume = {34}, number = {4}, year = {2017}, doi = {10.1016/j.anihpc.2016.05.006}, mrnumber = {3661864}, zbl = {1368.35045}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2016.05.006/} }
TY - JOUR AU - Caffarelli, L. AU - De Silva, D. AU - Savin, O. TI - The two membranes problem for different operators JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 899 EP - 932 VL - 34 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2016.05.006/ DO - 10.1016/j.anihpc.2016.05.006 LA - en ID - AIHPC_2017__34_4_899_0 ER -
%0 Journal Article %A Caffarelli, L. %A De Silva, D. %A Savin, O. %T The two membranes problem for different operators %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 899-932 %V 34 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2016.05.006/ %R 10.1016/j.anihpc.2016.05.006 %G en %F AIHPC_2017__34_4_899_0
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/
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