Boundary regularity for fully nonlinear integro-differential equations

Duke Mathematical Journal

Volumn 165, Issue 11, 2016, Pages 2079-2154

  Ros Oton, Xavier ^a   Serra, Joaquim ^b  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^b UNIVERSITAT POLITÈCNICA DE CATALUNYA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84962225696     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/00127094-3476700     Document Type: Article

References (51)

1. V. AKGIRAY and G. G. BOOTH, The stable-law model of stock returns, J. Bus. Econom. Statist. 6 (1988), 51-57. (2082)
2. G. BARLES, E. CHASSEIGNE, and C. IMBERT, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc. (JEMS) 13 (2011), 1-26. MR 2735074. DOI 10.4171/JEMS/242. (2085)
3. B. BASRAK, D. KRIZMANIC, and J. SEGERS, A functional limit theorem for partial sums of dependent random variables with infinite variance, Ann. Probab. 40 (2012), 2008-2033. MR 3025708. DOI 10.1214/11-AOP669. (2082)
4. R. BASS, Regularity results for stable-like operators, J. Funct. Anal. 257 (2009), 2693-2722. MR 2555009. DOI 10.1016/j.jfa.2009.05.012. (2146)
5. R. BASS and D. LEVIN, Harnack inequalities for jump processes, Potential Anal. 17 (2002), 375-388. MR 1918242. DOI 10.1023/A:1016378210944. (2085)
6. K. BOGDAN, The boundary Harnack principle for the fractional Laplacian, Studia Math. 123 (1997), 43-80. MR 1438304. (2080)
7. K. BOGDAN, T. KUMAGAI, and M. KWAŚNICKI, Boundary Harnack inequality for Markov processes with jumps, Trans. Amer. Math. Soc. 367 (2015), no. 1, 477-517. MR 3271268. DOI 10.1090/S0002-9947-2014-06127-8. (2082)
8. X. CABRé and Y. SIRE, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), 23-53. MR 3165278. DOI 10.1016/j.anihpc.2013.02.001. (2112)
9. L. CAFFARELLI, "Non-local diffusions, drifts and games" in Nonlinear Partial Differential Equations (Oslo, 2010), Abel Symp. 7, Springer, Heidelberg, 2012, 37-52. MR 3289358. DOI 10.1007/978-3-642-25361-4_3. (2084)
10. L. CAFFARELLI and X. CABRé, Fully Nonlinear Elliptic Equations, Amer. Math. Soc. Colloq. Publ. 43, Amer. Math. Soc., Providence, 1995. MR 1351007. (2083, 2087, 2146)
11. L. CAFFARELLI, J.-M. ROQUEJOFFRE, and Y. SIRE, Variational problems with free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), 1151-1179. MR 2677613. DOI 10.4171/JEMS/226. (2086)
12. L. CAFFARELLI, S. SALSA, and L. SILVESTRE, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math 171 (2008), 425-461. MR 2367025. DOI 10.1007/s00222-007-0086-6. (2086)
13. L. CAFFARELLI and L. SILVESTRE, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245-1260. MR 2354493. DOI 10.1080/03605300600987306. (2111, 2112, 2114)
14. L. CAFFARELLI, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), 597-638. MR 2494809. DOI 10.1002/cpa.20274. (2080, 2081, 2082, 2084, 2090, 2093, 2094, 2101, 2103, 2116, 2120, 2146)
15. L. CAFFARELLI, The Evans-Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math. (2) 174 (2011), 1163-1187. MR 2831115. DOI 10.4007/annals.2011.174.2.9. (2080, 2081, 2085)
16. L. CAFFARELLI, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal. 200 (2011), 59-88. MR 2781586. DOI 10.1007/s00205-010-0336-4. (2080, 2081, 2093, 2116, 2121)
17. L. CAFFARELLI, "Hölder regularity for generalized master equations with rough kernels" in Advances in Analysis: The Legacy of Elias M. Stein (Princeton, 2011), Princeton Univ. Press, Princeton, 2014, 63-83. MR 3329847. (2085)
18. H. CHANG LARA and G. DáVILA, Regularity for solutions of nonlocal, nonsymmetric equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), 833-859. MR 2995098. DOI 10.1016/j.anihpc.2012.04.006. (2085)
19. R. CONT and P. TANKOV, Financial Modelling With Jump Processes, Chapman & Hall, Boca Raton, 2004. MR 2042661. (2084)
20. G. GRUBB, Local and nonlocal boundary conditions for -transmission and fractional elliptic pseudodifferential operators, Anal. PDE 7 (2014), 1649-1682. MR 3293447. DOI 10.2140/apde.2014.7.1649. (2080)
21. G. GRUBB, Fractional Laplacians on domains, a development of Hörmander's theory of μ-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478-528. MR 3276603. DOI 10.1016/j.aim.2014.09.018. (2080, 2082, 2143)
22. N. GUILLEN and R. SCHWAB, Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations, Arch. Ration. Mech. Anal. 206 (2012), 111-157. MR 2968592. DOI 10.1007/s00205-012-0529-0. (2085)
23. C. HARDIN, G. SAMORODNITSKY and M. TAQQU, Nonlinear regression of stable random variables, Ann. Appl. Probab. 1 (1991), 582-612. MR 1129776. (2082)
24. M. KASSMANN and A. MIMICA, Intrinsic scaling properties for nonlocal operators, preprint, arXiv:1412.7566v3 [math. AP]. DOI 10.1016/j.spa.2012.09.016. (2085)
25. M. KASSMANN, M. RANG and R. SCHWAB, Integro-differential equations with nonlinear directional dependence, Indiana Univ. Math. J. 63 (2014), 1467-1498. MR 3283558. DOI 10.1512/iumj.2014.63.5394. (2085)
26. J. L. KAZDAN, Prescribing The Curvature Of A Riemannian Manifold, CBMS Reg. Conf. Ser. Math. 57, Amer. Math. Soc., Providence, 1985. MR 0787227. (2083, 2087, 2101, 2103)
27. T. J. KOZUBOWSKI, The theory of geometric stable distributions and its use in modeling financial data, Ph. D. dissertation, University of California at Santa Barbara, Santa Barbara, California, 1992. (2082)
28. 1, α interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Comm. Partial Differential Equations 38 (2013), 2081-2106. MR 3169771. DOI 10.1080/03605302.2013.831990. (2085)
29. N. KRYLOV, Boundedly inhomogeneous elliptic and parabolic equations in a domain (in Russian), Izv. Akad. Nauk SSSR Ser. Mat. 47 (1983), no. 1, 75-108; English translation in Math. USSR Izv. 22 (1984), no. 1, 67-98. MR 0688919. (2081, 2083)
30. N. N. LEBEDEV, Special Functions and their Applications, Dover, New York, 1972. MR 0350075. (2145)
31. P. LéVY, Théorie de l'addition des variables aléatoires, Gauthier-Villars, Paris, 1937. (2082)
32. S. MITTNIK and S. RACHEV, "Alternative multivariate stable distributions and their applications to financial modeling" in Stable Processes and Related Topics (Ithaca, N.Y., 1990), Progr. Probab. 25, Birkhäuser, Boston, 1991, 107-119. MR 1119354. (2082)
33. S. MITTNIK, Modeling asset returns with alternative stable distributions, Econometric Reviews 12 (1993), 261-330. MR 1249625. DOI 10.1080/07474939308800266. (2082)
34. J. P. NOLAN, "Multivariate stable distributions: Approximation, estimation, simulation and identification" in A Practical Guide to Heavy Tails (Santa Barbara, 1995), Birkhäuser, Boston, 1998, 509-525. MR 1652283. (2082)
35. J. P. NOLAN, "Fitting data and assessing goodness-of-fit with stable distributions" in Tails '99: Application of Heavy Tailed Distribution in Economics, Engineering and Statistics (Washington D.C., 1999), TSI Press, Albuquerque, 1999 (electronic CD). (2082)
36. J. P. NOLAN, Stable Distributions-Models for Heavy Tailed Data, preprint, http://academic2.american.edu/~jpnolan/stable/stable.html (accessed 3 February 2016). (2082)
37. J. P. NOLAN, A. K. PANORSKA and J. H. MCCULLOCH, "Estimation of stable spectral measures" in Stable Non-Gaussian Models in Finance and Econometrics, Elsevier, Oxford, 2001, 1113-1122. MR 1858839. DOI 10.1016/S0895-7177(01)00119-4. (2082)
38. B. OKSENDAL and A. SULEM, Applied Stochastic Control Of Jump Diffusions, Universitext, Springer, Berlin, 2005. MR 2109687. (2084)
39. A. K. PANORSKA, Generalized stable models for financial asset returns, J. Comput. Appl. Maht. 70 (1996), 111-114. DOI 10.1016/0377-0427(95)00144-1. (2082)
40. M. PIVATO and L. SECO, Estimating the spectral measure of a multivariate stable distribution via spherical harmonic analysis, J. Multivariate Anal. 87 (2003), 219-240. MR 2016936. DOI 10.1016/S0047-259X(03)00052-6. (2082)
41. N, Trans. Amer. Math. Soc. 146 (1969), 299-321. MR 0250372. (2082)
42. X. ROS-OTON and J. SERRA, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), 275-302. MR 3168912. DOI 10.1016/j.matpur.2013.06.003. (2080, 2083, 2091, 2114)
43. G. SAMORODNITSKY and M. S. TAQQU, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York, 1994. MR 1280932. (2082)
44. J. SERRA, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calc. Var. Partial Differential Equations 54 (2015), 615-629. MR 3385173. DOI 10.1007/s00526-014-0798-6. (2089)
45. σ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partial Differential Equations 54 (2015), 3571-3601. MR 3426087. DOI 10.1007/s00526-015-0914-2. (2085, 2089)
46. L. SILVESTRE, Hölder estimates for solutions of integro differential equations like the fractional Laplacian, Indiana Univ. Math. J. 55 (2006), 1155-1174. MR 2244602. DOI 10.1512/iumj.2006.55.2706. (2085)
47. L. SILVESTRE, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion, Adv. Math. 226 (2011), 2020-2039. MR 2737806. DOI 10.1016/j.aim.2010.09.007. (2085)
48. L. SILVESTRE and B. SIRAKOV, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations, Comm. Partial Differential Equations 39 (2014), 1694-1717. MR 3246039. DOI 10.1080/03605302.2013.842249. (2086, 2087)
49. P. SZTONYK, Boundary potential theory for stable Lévy processes, Colloq. Math. 95 (2003), 191-206. MR 1967420. DOI 10.4064/cm95-2-4. (2082)
50. P. SZTONYK, Regularity of harmonic functions for anisotropic fractional Laplacians, Math. Nachr. 283 (2010), 289-311. MR 2604123. DOI 10.1002/mana.200711116. (2082)
51. Z. X. WANG and D. R. GUO, Special Functions, World Scientific, Teaneck, N.J., 1989. MR 1034956. DOI 10.1142/0653. (2145)