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

François Hamel (1) , Xavier Ros-Oton (2) , Yannick Sire (1) , Enrico Valdinoci (3, 4)

1 I2M - Institut de Mathématiques de Marseille
2 University of Texas at Austin [Austin]
3 Università di Milano
4 WIAS - Weierstraß-Institut für Angewandte Analysis und Stochastik = Weierstrass Institute for Applied Analysis and Stochastics [Berlin]

We consider entire solutions to Lu = 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.

Mots clés: one-dimensional symmetry; stable solutions; De Giorgi Conjecture; Integral operators; convolution kernels; nonlocal equations

Domaines: Equations aux dérivées partielles [math.AP]

François Hamel, Xavier Ros-Oton, Yannick Sire, Enrico Valdinoci. A one-dimensional symmetry result for a class of nonlocal semilinear equations in the plane. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2017, 34 (2), pp.469-482. ⟨10.1016/j.anihpc.2016.01.001⟩. ⟨hal-01174183⟩