Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, p. 23-53
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This is the first of two articles dealing with the equation (-Δ) s v=f(v) in n , with s(0,1), where (-Δ) s stands for the fractional Laplacian — the infinitesimal generator of a Lévy process. This equation can be realized as a local linear degenerate elliptic equation in + n+1 together with a nonlinear Neumann boundary condition on + n+1 = n .In this first article, we establish necessary conditions on the nonlinearity f to admit certain type of solutions, with special interest in bounded increasing solutions in all of . These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian — in the spirit of a result of Modica for the Laplacian. Our proofs are uniform as s1, establishing in the limit the corresponding known results for the Laplacian.In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.

@article{AIHPC_2014__31_1_23_0,
     author = {Cabr\'e, Xavier and Sire, Yannick},
     title = {Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {31},
     number = {1},
     year = {2014},
     pages = {23-53},
     doi = {10.1016/j.anihpc.2013.02.001},
     zbl = {1286.35248},
     mrnumber = {3165278},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2014__31_1_23_0}
}
Cabré, Xavier; Sire, Yannick. Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 1, pp. 23-53. doi : 10.1016/j.anihpc.2013.02.001. http://www.numdam.org/item/AIHPC_2014__31_1_23_0/

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