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Boundary regularity for fully nonlinear integro-differential equations. (English) Zbl 1351.35245

L. Caffarelli and L. Silvestre [Commun. Pure Appl. Math. 62, No. 5, 597–638 (2009; Zbl 1170.45006)] studied the interior regularity for fully nonlinear nonlocal problems. In this important paper, the Authors analyse the case of the boundary regularity. They consider the class of nonlocal operators \({\mathcal L}_* \subset {\mathcal L}_o\) which consists of infinitesimal generators of stable Lévy processes belonging to the class \( {\mathcal L}_o\) introduced by Caffarelli and Silvestre. Consider the operator \(I\) elliptic with respect to \({\mathcal L}_* \). Let \(u\) be a the Dirichlet solution of \(Iu=f\) in \(\Omega\) with \(f\in C^\gamma\), then \(\frac{u}{d^s}\) is \(s+\gamma\) Hölder continuous in \(\bar{\Omega}\). The constants are stables when \(s \rightarrow 1\) and therefore the celebrated Krylov result is recovered by this approach.

MSC:

35R09 Integro-partial differential equations
45K05 Integro-partial differential equations

Citations:

Zbl 1170.45006
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References:

[1] V. Akgiray and G. G. Booth, The stable-law model of stock returns , J. Bus. Econom. Statist. 6 (1988), 51-57.
[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. · Zbl 1207.35277 · doi:10.4171/JEMS/242
[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. · Zbl 1295.60041 · doi:10.1214/11-AOP669
[4] R. Bass, Regularity results for stable-like operators , J. Funct. Anal. 257 (2009), 2693-2722. · Zbl 1177.45013 · doi:10.1016/j.jfa.2009.05.012
[5] R. Bass and D. Levin, Harnack inequalities for jump processes , Potential Anal. 17 (2002), 375-388. · Zbl 0997.60089 · doi:10.1023/A:1016378210944
[6] K. Bogdan, The boundary Harnack principle for the fractional Laplacian , Studia Math. 123 (1997), 43-80. · Zbl 0870.31009
[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. · Zbl 1309.60080 · doi:10.1090/S0002-9947-2014-06127-8
[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. · Zbl 1286.35248 · doi:10.1016/j.anihpc.2013.02.001
[9] L. Caffarelli, “Non-local diffusions, drifts and games” in Nonlinear Partial Differential Equations (Oslo, 2010) , Abel Symp. 7 , Springer, Heidelberg, 2012, 37-52. · Zbl 1266.35060 · doi:10.1007/978-3-642-25361-4_3
[10] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations , Amer. Math. Soc. Colloq. Publ. 43 , Amer. Math. Soc., Providence, 1995.
[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. · Zbl 1221.35453 · doi:10.4171/JEMS/226
[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. · Zbl 1148.35097 · doi:10.1007/s00222-007-0086-6
[13] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian , Comm. Partial Differential Equations 32 (2007), 1245-1260. · Zbl 1143.26002 · doi:10.1080/03605300600987306
[14] L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations , Comm. Pure Appl. Math. 62 (2009), 597-638. · Zbl 1170.45006 · doi:10.1002/cpa.20274
[15] L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations , Ann. of Math. (2) 174 (2011), 1163-1187. · Zbl 1232.49043 · doi:10.4007/annals.2011.174.2.9
[16] L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation , Arch. Ration. Mech. Anal. 200 (2011), 59-88. · Zbl 1231.35284 · doi:10.1007/s00205-010-0336-4
[17] L. Caffarelli and L. Silvestre, “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. · Zbl 1312.45013
[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. · Zbl 1317.35278 · doi:10.1016/j.anihpc.2012.04.006
[19] R. Cont and P. Tankov, Financial Modelling With Jump Processes , Chapman & Hall, Boca Raton, 2004. · Zbl 1052.91043 · doi:10.1201/9780203485217
[20] G. Grubb, Local and nonlocal boundary conditions for \(\mu\)-transmission and fractional elliptic pseudodifferential operators , Anal. PDE 7 (2014), 1649-1682. · Zbl 1317.35310 · doi:10.2140/apde.2014.7.1649
[21] G. Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of \(\mu\)-transmission pseudodifferential operators , Adv. Math. 268 (2015), 478-528. · Zbl 1318.47064 · doi:10.1016/j.aim.2014.09.018
[22] N. Guillen and R. Schwab, Aleksandrov-Bakelman-Pucci type estimates for integro-differential equations , Arch. Ration. Mech. Anal. 206 (2012), 111-157. · Zbl 1256.35190 · doi:10.1007/s00205-012-0529-0
[23] C. Hardin, G. Samorodnitsky and M. Taqqu, Nonlinear regression of stable random variables , Ann. Appl. Probab. 1 (1991), 582-612. · Zbl 0748.60017 · doi:10.1214/aoap/1177005840
[24] M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators , preprint, [math.AP]. arXiv:1412.7566v3 · Zbl 1371.35316
[25] M. Kassmann, M. Rang and R. Schwab, Integro-differential equations with nonlinear directional dependence , Indiana Univ. Math. J. 63 (2014), 1467-1498. · Zbl 1311.35047 · doi:10.1512/iumj.2014.63.5394
[26] J. L. Kazdan, Prescribing The Curvature Of A Riemannian Manifold , CBMS Reg. Conf. Ser. Math. 57 , Amer. Math. Soc., Providence, 1985. · Zbl 0561.53048
[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.
[28] D. Kriventsov, \(C^{1,\alpha}\) interior regularity for nonlinear nonlocal elliptic equations with rough kernels , Comm. Partial Differential Equations 38 (2013), 2081-2106. · Zbl 1281.35092 · doi:10.1080/03605302.2013.831990
[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.
[30] N. N. Lebedev, Special Functions and their Applications , Dover, New York, 1972. · Zbl 0271.33001
[31] P. Lévy, Théorie de l’addition des variables aléatoires , Gauthier-Villars, Paris, 1937.
[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. · Zbl 0725.90006 · doi:10.1007/978-1-4684-6778-9_6
[33] S. Mittnik and S. Rachev, Modeling asset returns with alternative stable distributions , Econometric Reviews 12 (1993), 261-330. · Zbl 0801.62096 · doi:10.1080/07474939308800266
[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.
[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).
[36] J. P. Nolan, Stable Distributions-Models for Heavy Tailed Data , preprint, (accessed 3 February 2016).
[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. · Zbl 1004.62028 · doi:10.1016/S0895-7177(01)00119-4
[38] B. Oksendal and A. Sulem, Applied Stochastic Control Of Jump Diffusions , Universitext, Springer, Berlin, 2005.
[39] A. K. Panorska, Generalized stable models for financial asset returns , J. Comput. Appl. Maht. 70 (1996), 111-114. · Zbl 0852.90017 · doi:10.1016/0377-0427(95)00144-1
[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. · Zbl 1041.60019 · doi:10.1016/S0047-259X(03)00052-6
[41] W. E. Pruitt and S. J. Taylor, The potential kernel and hitting probabilities for the general stable process in \(\mathbb{R}^{N}\) , Trans. Amer. Math. Soc. 146 (1969), 299-321. · Zbl 0229.60052
[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. · Zbl 1285.35020 · doi:10.1016/j.matpur.2013.06.003
[43] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance , Chapman and Hall, New York, 1994. · Zbl 0925.60027
[44] J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels , Calc. Var. Partial Differential Equations 54 (2015), 615-629. · Zbl 1327.35170 · doi:10.1007/s00526-014-0798-6
[45] J. Serra, \(C^{\sigma+\alpha}\) regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels , Calc. Var. Partial Differential Equations 54 (2015), 3571-3601. · Zbl 1344.35042 · doi:10.1007/s00526-015-0914-2
[46] L. Silvestre, Hölder estimates for solutions of integro differential equations like the fractional Laplacian , Indiana Univ. Math. J. 55 (2006), 1155-1174. · Zbl 1101.45004 · doi:10.1512/iumj.2006.55.2706
[47] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion , Adv. Math. 226 (2011), 2020-2039. · Zbl 1216.35165 · doi:10.1016/j.aim.2010.09.007
[48] L. Silvestre and B. Sirakov, Boundary regularity for viscosity solutions of fully nonlinear elliptic equations , Comm. Partial Differential Equations 39 (2014), 1694-1717. · Zbl 1308.35043 · doi:10.1080/03605302.2013.842249
[49] P. Sztonyk, Boundary potential theory for stable Lévy processes , Colloq. Math. 95 (2003), 191-206. · Zbl 1026.60093 · doi:10.4064/cm95-2-4
[50] P. Sztonyk, Regularity of harmonic functions for anisotropic fractional Laplacians , Math. Nachr. 283 (2010), 289-311. · Zbl 1194.47044 · doi:10.1002/mana.200711116
[51] Z. X. Wang and D. R. Guo, Special Functions , World Scientific, Teaneck, N.J., 1989.
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