Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates

Cabré, Xavier; Sire, Yannick

Annales de l'Institut Henri Poincaré. C, Analyse non linéaire, Tome 31 (2014) no. 1, pp. 23-53

This is the first of two articles dealing with the equation ${(-\Delta )}^{s}v=f\left(v\right)$ in ${ℝ}^n$, with $s\in (0,1)$, where ${(-\Delta )}^s$ 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 ${ℝ}_{+}^{n+1}$ together with a nonlinear Neumann boundary condition on $\partial {ℝ}_{+}^{n+1}={ℝ}^n$.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 $ℝ$. 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 $s\uparrow 1$, 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.

[1] L. Ambrosio, X. Cabré, Entire solutions of semilinear elliptic equations in ${ℝ}^3$ 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 | MR | EuDML | 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

[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 | MR | EuDML | 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 | MR | EuDML | Zbl | Numdam

[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 ${(-\delta )}^{s}q+q-q^{\alpha +1}=0$ in $ℝ$, 2010) | arXiv

[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 $W^{1,p}(\Omega ,d_M^{ϵ})$ on M, Czechoslovak Math. J. 43 no. 4 (1993), 695-711 | MR | EuDML | 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