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.

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.

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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://archive.numdam.org/articles/10.1016/j.anihpc.2013.02.001/

[1] L. Ambrosio, X. Cabré, Entire solutions of semilinear elliptic equations in 3 and a conjecture of De Giorgi, J. Amer. Math. Soc. 13 no. 4 (2000), 725-739 | MR | Zbl

[2] K. Astala, L. Päivärinta, A boundary integral equation for Calderón's inverse conductivity problem, Collect. Math. Vol. Extra (2006), 127-139 | EuDML | MR | Zbl

[3] J. Bertoin, 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] X. Cabré, J. Solà-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 no. 12 (2005), 1678-1732 | MR | Zbl

[6] L. Caffarelli, A. Mellet, Y. Sire, Traveling waves for a boundary reaction–diffusion equation, Adv. Math. 230 no. 2 (2012), 433-457 | MR | Zbl

[7] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 no. 8 (2007), 1245 | MR | Zbl

[8] L.A. Caffarelli, J.-M. Roquejoffre, Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 no. 5 (2010), 1151-1179 | EuDML | MR | Zbl

[9] P.R. Chernoff, J.E. Marsden, Properties of Infinite Dimensional Hamiltonian Systems, Lecture Notes in Math. vol. 425, Springer-Verlag, Berlin (1974) | MR | Zbl

[10] E. Fabes, D. Jerison, C. Kenig, The Wiener test for degenerate elliptic equations, Ann. Inst. Fourier (Grenoble) 32 no. 3 (1982), 151-182 | EuDML | Numdam | MR | Zbl

[11] E.B. Fabes, C.E. Kenig, R.P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 no. 1 (1982), 77-116 | MR | Zbl

[12] R.L. Frank, E. Lenzmann, Uniqueness and nondegeneracy of ground states for (-δ) s q+q-q α+1 =0 in , arXiv:1009.4042 (2010)

[13] A. Garroni, S. Müller, Γ-limit of a phase-field model of dislocations, SIAM J. Math. Anal. 36 no. 6 (2005), 1943-1964 | MR | Zbl

[14] C. Imbert, R. Monneau, Homogenization of first-order equations with (u/ϵ)-periodic Hamiltonians, I. Local equations, Arch. Ration. Mech. Anal. 187 no. 1 (2008), 49-89 | MR | Zbl

[15] N.S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. vol. 180, Springer-Verlag, New York (1972) | MR | Zbl

[16] R. Mancinelli, D. Vergni, A. Vulpiani, Front propagation in reactive systems with anomalous diffusion, Phys. D 185 no. 3–4 (2003), 175-195 | MR | Zbl

[17] L. Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations, Comm. Pure Appl. Math. 38 no. 5 (1985), 679-684 | MR | Zbl

[18] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226 | MR | Zbl

[19] B. Muckenhoupt, E.M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17-92 | MR | Zbl

[20] A. Nekvinda, Characterization of traces of the weighted Sobolev space W 1,p (Ω,d M ϵ ) on M, Czechoslovak Math. J. 43 no. 4 (1993), 695-711 | EuDML | MR | Zbl

[21] L. Silvestre, 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] P.R. Stinga, J.L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations 35 no. 11 (2010), 2092-2122 | MR | Zbl

[23] J. Tan, J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst. 31 no. 3 (2011), 975-983 | MR | Zbl

[24] J.F. Toland, The Peierls–Nabarro and Benjamin–Ono equations, J. Funct. Anal. 145 no. 1 (1997), 136-150 | MR | Zbl

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