Regularity results for fully nonlinear parabolic integro-differential operators. (English) Zbl 1283.47054
The authors consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. There are some known results about Harnack inequalities and Hölder regularity for integro-differential operators with symmetric and nonsymmetric kernels. However, such kind of results cannot be easily generalized in the parabolic setting. The difficulties arise from the fact that the equation is local in time while it is nonlocal in the space variables and the classical approach due to Caffarelli-Silvestre cannot be easily applied in this situation. The authors are able to handle these difficulties first proving nonlocal versions of the parabolic nonlocal Alexandroff-Backelman-Pucci estimate. Then they construct a special function and obtain the decay estimates of upper level sets. Lastly, they use these results to prove Hölder regularity, interior \(C^{1,\alpha}\) -regularity and Harnack estimates.
Reviewer: Vincenzo Vespri (Firenze)
MSC:
| 47G20 | Integro-differential operators |
| 45K05 | Integro-partial differential equations |
| 35J60 | Nonlinear elliptic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 60J75 | Jump processes (MSC2010) |
Keywords:
fully nonlinear parabolic integro-differential equations; Harnack inequality; Hölder regularity; \(C^{1,\alpha}\)-regularityCitations:
Zbl 1170.45006References:
| [1] | Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linaire 13, 293-317 (1996) · Zbl 0870.45002 |
| [2] | Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60(1-2), 57-83 (1997) · Zbl 0878.60036 |
| [3] | Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Preprint. SIAM J. Control Optim. 49(3), 948-962 (2011) · Zbl 1155.45004 |
| [4] | Barlow, M.T., Bass, R.F.. Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361(4), 1963-1999 (2009) · Zbl 1166.60045 |
| [5] | Bass, R.F., Kassmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Am. Math. Soc. 357(2), 837-850 (2005) · Zbl 1052.60060 · doi:10.1090/S0002-9947-04-03549-4 |
| [6] | Bass, R.F., Kassmann, M.: Hölder continuity of harmonic functions with respect to operators of variable order. Commun. Partial Differ. Equ. 30(7-9), 1249-1259 (2005) · Zbl 1087.45004 · doi:10.1080/03605300500257677 |
| [7] | Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potential Anal. 17(4), 375-388 (2002) · Zbl 0997.60089 |
| [8] | Bensaoud, I., Sayah, A.: Stability results for Hamilton-Jacobi equations with integro-differential terms and discontinuous Hamiltonians. Arch. Math. 79, 392-395 (2002) · Zbl 1022.35004 · doi:10.1007/PL00012462 |
| [9] | Caffarelli, L.A.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. (2) 130(1), 189-213 (1989) · Zbl 0692.35017 |
| [10] | Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geometrophic equation. Ann. Math. 171(3), 1903-1930 (2010) · Zbl 1204.35063 |
| [11] | Caffarelli, L.A., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597-638 (2009) · Zbl 1170.45006 · doi:10.1002/cpa.20274 |
| [12] | Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society, vol. 43. Colloquium Publications, American Mathematical Society, Providence (1995) · Zbl 0834.35002 |
| [13] | Dong, G.: Nonlinear Partial Differential Equations of Second Order, Translations of Mathematical Monographs, vol. 95. American Mathematical Society, Providence (1991) · Zbl 0759.35001 |
| [14] | Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224. Springer, Berlin (1983) · Zbl 0562.35001 |
| [15] | De Giorgi, E., Dal Maso, G.: \[ \Gamma\] Γ-Convergence and Calculus of Variations. Lecture Notes in Mathematics, vol. 979. Springer, Berlin (1983) · Zbl 0511.49007 |
| [16] | Ishii, H.: On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs. Commun. Pure Appl. Math. 42(1), 15-45 (1989) · Zbl 0645.35025 · doi:10.1002/cpa.3160420103 |
| [17] | Jakobsen, E.R., Karlsen, K.H.: A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations. Nonlinear Differ. Equ. Appl. 13, 137-165 (2006) · Zbl 1105.45006 · doi:10.1007/s00030-005-0031-6 |
| [18] | Jensen, R.: The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations. Arch. Ration. Mech. Anal. 101(1), 1-27 (1988) · Zbl 0708.35019 · doi:10.1007/BF00281780 |
| [19] | Kim, Y.-C., Lee, K.-A.: Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels: subcritical Case. Potential Anal. 38(2), 433-455 (2013) · Zbl 1263.47060 |
| [20] | Kim, Y.-C., Lee, K.-A.: Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels. Manuscr. Math. 139, 291-319 (2012) · Zbl 1258.47064 |
| [21] | Krylov, N.V., Safonov, M.V.: An estimate for the probability of a diffusion process hitting a set of positive measure. Dokl. Akad. Nauk. SSSR 245, 18-20 (1979) |
| [22] | Pham, H.: Optimal stopping of controlled jump-diffusion processes: a viscosity solutions approach. J. Math. Syst. Estim. Control 8(1), 1-27 (1998) |
| [23] | Sayah, A.: Équations d’Hamilton-Jacobi du premier ordre avec termes intégro-différentiels. I Unicité des solutions de viscosité. Commun. Partial Differ. Equ. 16(6-7), 1057-1074 (1991) · Zbl 0742.45004 |
| [24] | Silvestre, L.: Hölder estimates for solutions of integro-differential equations like the fractional laplace. Indiana Univ. Math. J. 55(3), 1155-1174 (2006) · Zbl 1101.45004 |
| [25] | Soner, H.M.: Optimal control with state-space constraint. II. SIAM J. Control Optim. 24(6), 1110-1122 (1986) · Zbl 0619.49013 |
| [26] | Song, R., Vondraček, Z.: Harnack inequality for some classes of Markov processes. Math. Z. 246(1-2), 177-202 (2004) · Zbl 1052.60064 |
| [27] | Tso, K.: Deforming a hypersurface by its Gauss-Kronecker curvature. Commun. Pure Appl. Math. 38, 867-882 (1985) · Zbl 0612.53005 · doi:10.1002/cpa.3160380615 |
| [28] | Wang, L.: On the regularity theory of fully nonlinear parabolic equations. I. Commun. Pure Appl. Math. 45(1), 2776 (1992) · Zbl 0832.35025 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
