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Regularity results and large time behavior for integro-differential equations with coercive Hamiltonians. (English) Zbl 1336.35347

From the author’s abstract: We obtain regularity results for elliptic integro-differential equations driven by the stronger effect of coercive gradient terms. This feature allows us to construct suitable strict supersolutions from which we conclude Holder estimates for bounded subsolutions.

MSC:

35R09 Integro-partial differential equations
35B51 Comparison principles in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35D40 Viscosity solutions to PDEs
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Annales de L’I.H.P., section C 13(3), 293-317 (1996) · Zbl 0870.45002
[2] Bardi, M., Da Lio, F.: On the strong maximum principle for fully nonlinear degenerate elliptic equations. Arch. Math. (Basel) 73(4), 276-285 (1999) · Zbl 0939.35038 · doi:10.1007/s000130050399
[3] Barles, G.: Solutions de Viscosite des Equations de Hamilton-Jacobi Collection “Mathematiques et Applications” de la SIAM, no 17. Springer (1994) · Zbl 0819.35002
[4] Barles, G.: A short proof of the \[C^{0,\alpha }-\] C0,α- regularity of viscosity subsolutions for superquadratic viscous Hamilton-Jacobi equations and applications. Nonlinear Anal. 73, 31-47 (2010) · Zbl 1190.35081 · doi:10.1016/j.na.2010.02.009
[5] Barles, G., Chasseigne, E., Ciomaga, A., Imbert, C.: Lipschitz regularity of solutions for mixed integro-differential equations. J. Differ. Equ. 252, 6012-6060 (2012) · Zbl 1298.35033 · doi:10.1016/j.jde.2012.02.013
[6] Barles, G., Chasseigne, E., Ciomaga, A., Imbert, C.: Time behavior of periodic viscosity solutions for uniformly parabolic integro-differential equations. Calc. Var. Partial Differ. Equ. 50(1-2), 283-304 (2014) · Zbl 1293.35037 · doi:10.1007/s00526-013-0636-2
[7] Barles, G., Chasseigne, E., Georgelin, C., Jakobsen, E.: On Neumann type problems for nonlocal equations set in a half space. Trans. Am. Math. Soc. 366(9), 4873-4917 (2014) · Zbl 1323.35192 · doi:10.1090/S0002-9947-2014-06181-3
[8] Barles, G., Chasseigne, E., Imbert, C.: On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57(1), 213-246 (2008) · Zbl 1139.47057
[9] Barles, G., Imbert, C.: Second-order eliptic integro-differential equations: viscosity solutions’ theory revisited. IHP Anal. Non Linéare 25(3), 567-585 (2008) · Zbl 1155.45004
[10] Barles, G., Souganidis, P.E.: Space-time periodic solutions and long-time behavior of solutions of quasilinear parabolic equations. SIAM J. Math. Anal. 32, 1311-1323 (2001) · Zbl 0986.35047 · doi:10.1137/S0036141000369344
[11] Bogdan, K., Burdzy, K., Chen, Z.-Q.: Censored stable processes. Probab. Theory Rel. Fields 127(1), 89-152 (2003) · Zbl 1032.60047 · doi:10.1007/s00440-003-0275-1
[12] Caffarelli, L., Silvestre, L.: Regularity theory for nonlocal integro-differential equations. Commun. Pure Appl. Math. 62(5), 597-638 (2009) · Zbl 1170.45006 · doi:10.1002/cpa.20274
[13] Capuzzo-Dolcetta, I., Leoni, F., Porretta, A.: Hölder estimates for degenerate elliptic equations with coercive Hamiltonians. Trans. Am. Math. Soc. 362(9), 4511-4536 (2010) · Zbl 1198.35110 · doi:10.1090/S0002-9947-10-04807-5
[14] Cardaliaguet, P., Cannarsa, P.: Hölder estimates in space-time for viscosity solutions of Hamilton-Jacobi equations. CPAM 63(5), 590-629 (2010) · Zbl 1206.35052
[15] Cardaliaguet, P., Rainer, C.: Höder regularity for viscosity solutions of fully nonlinear, local or nonlocal, Hamilton-Jacobi equations with superquadratic growth in the gradient. SIAM J. Control Optim. 49(2), 555-573 (2011) · Zbl 1218.49044 · doi:10.1137/100784400
[16] Chasseigne, E.: The Dirichlet problem for some nonlocal diffusion equations. Differ. Integr. Equ. 20(12), 1389-1404 (2007) · Zbl 1211.47088
[17] Ciomaga, A.: On the strong maximum principle for second order nonlinear parabolic integro-differential equations. Adv. Differ. Equ. 17, 635-671 (2012) · Zbl 1264.35262
[18] Coville, J.: Maximum principles, sliding techniques and applications to nonlocal equations. Electron. J. Differ. Equ. 2007(68), 1-23 (2007) · Zbl 1137.35321
[19] Coville, J.: Remarks on the strong maximum principle for nonlocal operators. Electron. J. Differ. Equ. 66, 1-10 (2008) · Zbl 1173.35390
[20] Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1-67 (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5
[21] Da Lio, F.: Comparison results for quasilinear equations in annular domains and applications. Commun. Partial Differ. Equ. 27(1 & 2), 283-323 (2002) · Zbl 0994.35014 · doi:10.1081/PDE-120002788
[22] Di Neza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521-573 (2012) · Zbl 1252.46023 · doi:10.1016/j.bulsci.2011.12.004
[23] Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes. de Grueter Stud. Math. 19 (1994) · Zbl 0838.31001
[24] Guan, Q.Y.: The integration by part of the regional fractional Laplacian. Commun. Math. Phys. 266, 289-329 (2006) · Zbl 1121.60051 · doi:10.1007/s00220-006-0054-9
[25] Guan, Q.Y., Ma, Z.M.: Reflected symmetric \[\alpha\] α stable process and regional fractional Laplacian. Probab. Theory Relat. Fields 134, 649-694 (2006) · Zbl 1089.60030 · doi:10.1007/s00440-005-0438-3
[26] Ishii, H.: Existence and uniqueness of solutions of Hamilton-Jacobi equations. Funkcialaj Ekvacioj 29, 167-188 (1986) · Zbl 0614.35011
[27] Ishii, H., Nakamura, G.: A class of integral equations and approximation of p-Laplace equations. Calc. Var. Partial Differ. Equ. 37(3-4), 485-522 (2010) · Zbl 1198.45005
[28] Jacob, N.: Pseudo Differential Operators and Markov Process. Markov Process and Applications, vol. III. Imperial College Press, Princeton (2005) · doi:10.1142/9781860947155
[29] Kim, P.: Weak convergence of censored and reflected stable processes. Stoch. Process. Appl. 116(12), 1792-1814 (2006) · Zbl 1105.60007 · doi:10.1016/j.spa.2006.04.006
[30] Koike, S.: A beginner’s guide to the theory of viscosity solutions. MSJ Memoirs, vol. 13. Mathematical Society of Japan, Tokyo (2004) · Zbl 1056.49027
[31] Sayah, A.: Équations d’Hamilton-Jacobi du emier Ordre Avec Termes Intégro-Différentiels. I. Unicité des solutions de viscosité. Commun. Partial Differ. Equ. 16, 1057-1074 (1991) · Zbl 0742.45004 · doi:10.1080/03605309108820789
[32] Sayah, A.: Équations d’Hamilton-Jacobi du emier Ordre Avec Termes Intégro-Différentiels. II. Existence de solutions de viscosité. Commun. Partial Differ. Equ. 16, 1075-1093 (1991) · Zbl 0742.45005 · doi:10.1080/03605309108820790
[33] Tchamba, T.T.: Large time behavior of solutions of viscous Hamilton-Jacobi equations with superquadratic Hamiltonian. Asymptot. Anal. 66, 161-186 (2010) · Zbl 1195.35183
[34] Topp, E.: Existence and uniqueness for integro-differential equations with dominating drift terms. Commun. Partial Differ. Equ. 39(8), 1523-1554 (2014) · Zbl 1326.47056 · doi:10.1080/03605302.2014.900567
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