Geodesic
A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 469-482

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We consider entire solutions to Lu=f(u) in R2, where L is a nonlocal operator with translation invariant, even and compactly supported kernel K. Under different assumptions on the operator L, we show that monotone solutions are necessarily one-dimensional. The proof is based on a Liouville type approach. A variational characterization of the stability notion is also given, extending our results in some cases to stable solutions.

DOI : 10.1016/j.anihpc.2016.01.001
Classification : 45A05, 47G10, 47B34, 35R11
Keywords: Integral operators, Convolution kernels, Nonlocal equations, Stable solutions, One-dimensional symmetry, De Giorgi Conjecture 
@article{AIHPC_2017__34_2_469_0,
     author = {Hamel, Fran\c{c}ois and Ros-Oton, Xavier and Sire, Yannick and Valdinoci, Enrico},
     title = {A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {469--482},
     publisher = {Elsevier},
     volume = {34},
     number = {2},
     year = {2017},
     doi = {10.1016/j.anihpc.2016.01.001},
     mrnumber = {3610941},
     zbl = {1358.45004},
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     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.01.001/}
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Hamel, François; Ros-Oton, Xavier; Sire, Yannick; Valdinoci, Enrico. A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 469-482. doi: 10.1016/j.anihpc.2016.01.001

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