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

Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire

Volumn 31, Issue 1, 2014, Pages 23-53

  Cabré, Xavier ^a   Sire, Yannick ^b  

^a UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

^b AIX MARSEILLE UNIVERSITY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; HAMILTONIANS; LAPLACE TRANSFORMS; NONLINEAR EQUATIONS;

FRACTIONAL LAPLACIAN; INCREASING SOLUTIONS; INFINITESIMAL GENERATOR; LAPLACIANS; LOCAL LINEAR; NEUMANN BOUNDARY CONDITION;

CONTROL NONLINEARITIES;

EID: 84894039977     PISSN: 02941449     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.anihpc.2013.02.001     Document Type: Article

References (24)

1. R 3 and a conjecture of De Giorgi J. Amer. Math. Soc. 13 4 2000 725 739
2. K. Astala, and L. Päivärinta A boundary integral equation for Calderón's inverse conductivity problem Collect. Math. Vol. Extra 2006 127 139
3. J. Bertoin Lévy Processes Cambridge Tracts in Math. vol. 121 1996 Cambridge University Press Cambridge
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, and J. Solà-Morales Layer solutions in a half-space for boundary reactions Comm. Pure Appl. Math. 58 12 2005 1678 1732
6. L. Caffarelli, A. Mellet, and Y. Sire Traveling waves for a boundary reaction-diffusion equation Adv. Math. 230 2 2012 433 457
7. L. Caffarelli, and L. Silvestre An extension problem related to the fractional Laplacian Comm. Partial Differential Equations 32 8 2007 1245
8. L.A. Caffarelli, J.-M. Roquejoffre, and Y. Sire Variational problems for free boundaries for the fractional Laplacian J. Eur. Math. Soc. (JEMS) 12 5 2010 1151 1179
9. P.R. Chernoff, and J.E. Marsden Properties of Infinite Dimensional Hamiltonian Systems Lecture Notes in Math. vol. 425 1974 Springer-Verlag Berlin
10. E. Fabes, D. Jerison, and C. Kenig The Wiener test for degenerate elliptic equations Ann. Inst. Fourier (Grenoble) 32 3 1982 151 182
11. E.B. Fabes, C.E. Kenig, and R.P. Serapioni The local regularity of solutions of degenerate elliptic equations Comm. Partial Differential Equations 7 1 1982 77 116
12. 1 = 0 in R preprint arXiv:1009.4042 2010
13. A. Garroni, and S. Müller Γ-limit of a phase-field model of dislocations SIAM J. Math. Anal. 36 6 2005 1943 1964 (electronic)
14. C. Imbert, and R. Monneau Homogenization of first-order equations with (u / Ïμ) -periodic Hamiltonians, I. Local equations Arch. Ration. Mech. Anal. 187 1 2008 49 89
15. N.S. Landkof Foundations of Modern Potential Theory Grundlehren Math. Wiss. vol. 180 1972 Springer-Verlag New York translated from the Russian by A.P. Doohovskoy
16. R. Mancinelli, D. Vergni, and A. Vulpiani Front propagation in reactive systems with anomalous diffusion Phys. D 185 3-4 2003 175 195
17. L. Modica A gradient bound and a Liouville theorem for nonlinear Poisson equations Comm. Pure Appl. Math. 38 5 1985 679 684
18. B. Muckenhoupt Weighted norm inequalities for the Hardy maximal function Trans. Amer. Math. Soc. 165 1972 207 226
19. B. Muckenhoupt, and E.M. Stein Classical expansions and their relation to conjugate harmonic functions Trans. Amer. Math. Soc. 118 1965 17 92
20. , p (Ω, dMε) on M Czechoslovak Math. J. 43 4 1993 695 711
21. L. Silvestre Regularity of the obstacle problem for a fractional power of the Laplace operator Comm. Pure Appl. Math. 60 1 2007 67 112
22. P.R. Stinga, and J.L. Torrea Extension problem and Harnack's inequality for some fractional operators Comm. Partial Differential Equations 35 11 2010 2092 2122
23. J. Tan, and J. Xiong A Harnack inequality for fractional Laplace equations with lower order terms Discrete Contin. Dyn. Syst. 31 3 2011 975 983
24. J.F. Toland The Peierls-Nabarro and Benjamin-Ono equations J. Funct. Anal. 145 1 1997 136 150