%0 Unpublished work
%T Entire solutions to semilinear nonlocal equations in $R^2$
%+ University of Texas at Austin [Austin]
%+ Institut de Mathématiques de Marseille (I2M)
%A Ros-Oton, Xavier
%A Sire, Yannick
%Z X.R. was supported by grants MTM2011-27739-C04-01 (Spain), and 2009SGR345 (Catalunya).
%8 2015-05-26
%D 2015
%Z 1505.06919
%Z Mathematics [math]/Analysis of PDEs [math.AP]Preprints, Working Papers, ...
%X We consider entire solutions to $L u= f(u)$ in $R^2$, where $L$ is a general nonlocal operator with kernel $K(y)$. Under certain natural assumtions on the operator $L$, we show that any stable solution is a 1D solution. In particular, our result applies to any solution $u$ which is monotone in one direction. Compared to other proofs of the De Giorgi type results on nonlocal equations, our method is the first successfull attempt to use the Liouville theorem approach to get flatness of the level sets.
%G English
%L hal-01341881
%U https://hal.science/hal-01341881
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ OPENAIRE
%~ I2M
%~ I2M-2014-
%~ TDS-MACS
%~ ANR