A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane

Hamel, François; Ros-Oton, Xavier; Sire, Yannick; Valdinoci, Enrico

doi:10.1016/j.anihpc.2016.01.001

Hamel, François ; Ros-Oton, Xavier ; Sire, Yannick ; Valdinoci, Enrico

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 2, pp. 469-482.

Résumé

We consider entire solutions to $L u = f (u)$ in $R^{2}$ , 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.

MR Zbl

DOI : 10.1016/j.anihpc.2016.01.001

Classification : 45A05, 47G10, 47B34, 35R11
Mots clés : Integral operators, Convolution kernels, Nonlocal equations, Stable solutions, One-dimensional symmetry, De Giorgi Conjecture

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     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},
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     url = {http://archive.numdam.org/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. http://archive.numdam.org/articles/10.1016/j.anihpc.2016.01.001/

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