This is the first of two articles dealing with the equation in , with , where 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 together with a nonlinear Neumann boundary condition on .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 , 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}, pages = {23--53}, publisher = {Elsevier}, volume = {31}, number = {1}, year = {2014}, doi = {10.1016/j.anihpc.2013.02.001}, mrnumber = {3165278}, zbl = {1286.35248}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.001/} }
TY - JOUR AU - Cabré, Xavier AU - Sire, Yannick TI - Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 23 EP - 53 VL - 31 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.001/ DO - 10.1016/j.anihpc.2013.02.001 LA - en ID - AIHPC_2014__31_1_23_0 ER -
%0 Journal Article %A Cabré, Xavier %A Sire, Yannick %T Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 23-53 %V 31 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.02.001/ %R 10.1016/j.anihpc.2013.02.001 %G en %F 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/articles/10.1016/j.anihpc.2013.02.001/
[1] Entire solutions of semilinear elliptic equations in and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 no. 4 (2000), 725-739 | MR | Zbl
, ,[2] A boundary integral equation for Calderón's inverse conductivity problem, Collect. Math. Vol. Extra (2006), 127-139 | EuDML | MR | Zbl
, ,[3] Lévy Processes, Cambridge Tracts in Math. vol. 121, Cambridge University Press, Cambridge (1996) | MR | Zbl
,[4] X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians, II: existence, uniqueness, and qualitative properties of solutions, arXiv:1111.0796v1, 2011; Trans. Amer. Math. Soc., in press.
[5] Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 no. 12 (2005), 1678-1732 | MR | Zbl
, ,[6] Traveling waves for a boundary reaction–diffusion equation, Adv. Math. 230 no. 2 (2012), 433-457 | MR | Zbl
, , ,[7] An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 no. 8 (2007), 1245 | MR | Zbl
, ,[8] Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 no. 5 (2010), 1151-1179 | EuDML | MR | Zbl
, , ,[9] Properties of Infinite Dimensional Hamiltonian Systems, Lecture Notes in Math. vol. 425, Springer-Verlag, Berlin (1974) | MR | Zbl
, ,[10] The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 no. 3 (1982), 151-182 | EuDML | Numdam | MR | Zbl
, , ,[11] The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 no. 1 (1982), 77-116 | MR | Zbl
, , ,[12] Uniqueness and nondegeneracy of ground states for in , arXiv:1009.4042 (2010)
, ,[13] Γ-limit of a phase-field model of dislocations, SIAM J. Math. Anal. 36 no. 6 (2005), 1943-1964 | MR | Zbl
, ,[14] Homogenization of first-order equations with -periodic Hamiltonians, I. Local equations, Arch. Ration. Mech. Anal. 187 no. 1 (2008), 49-89 | MR | Zbl
, ,[15] Foundations of Modern Potential Theory, Grundlehren Math. Wiss. vol. 180, Springer-Verlag, New York (1972) | MR | Zbl
,[16] Front propagation in reactive systems with anomalous diffusion, Phys. D 185 no. 3–4 (2003), 175-195 | MR | Zbl
, , ,[17] A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 no. 5 (1985), 679-684 | MR | Zbl
,[18] Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226 | MR | Zbl
,[19] Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17-92 | MR | Zbl
, ,[20] Characterization of traces of the weighted Sobolev space on M, Czechoslovak Math. J. 43 no. 4 (1993), 695-711 | EuDML | MR | Zbl
,[21] Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 no. 1 (2007), 67-112 | MR | Zbl
,[22] Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations 35 no. 11 (2010), 2092-2122 | MR | Zbl
, ,[23] A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst. 31 no. 3 (2011), 975-983 | MR | Zbl
, ,[24] The Peierls–Nabarro and Benjamin–Ono equations, J. Funct. Anal. 145 no. 1 (1997), 136-150 | MR | Zbl
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