On the differentiability of the solutions of non-local Isaacs equations involving \(\frac{1}{2}\)-Laplacian. (English) Zbl 1334.35381
Summary: We derive \(C^{1,\sigma}\)-estimate for the solutions of a class of non-local elliptic Bellman-Isaacs equations. These equations are fully nonlinear and are associated with infinite horizon stochastic differential game problems involving jump-diffusions. The non-locality is represented by the presence of fractional order diffusion term and we deal with the particular case of \(\frac 12\)-Laplacian, where the order \(\frac 12\) is known as the critical order in this context. More importantly, these equations are not translation invariant and we prove that the viscosity solutions of such equations are \(C^{1,\sigma}\), making the equations classically solvable.
MSC:
| 35R11 | Fractional partial differential equations |
| 35F21 | Hamilton-Jacobi equations |
| 45K05 | Integro-partial differential equations |
| 49L20 | Dynamic programming in optimal control and differential games |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 91A23 | Differential games (aspects of game theory) |
| 35R09 | Integro-partial differential equations |
Keywords:
viscosity solution; differentiability; Bellman-Isaacs equations; integro-partial differential equation; fractional LaplacianReferences:
| [1] | G. Barles, On the Dirichlet problem for second-order elliptic integro-differential equations,, Indiana Univ. Math. J., 57, 213 (2008) · Zbl 1139.47057 · doi:10.1512/iumj.2008.57.3315 |
| [2] | G. Barles, Second-order elliptic integro-differential equations: Viscosity solutions’ theory revisited,, Ann. Inst. H. Poincare Anal. Non Linaire, 25, 567 (2008) · Zbl 1155.45004 · doi:10.1016/j.anihpc.2007.02.007 |
| [3] | I. H. Biswas, On zero-sum stochastic differential games with jump-diffusion driven state: a viscosity solution framework,, SIAM J. Control Optim., 50, 1823 (2012) · Zbl 1253.91027 · doi:10.1137/080720504 |
| [4] | I. H. Biswas, Regularization by \(\frac{1}{2} \)-Laplacian and vanishing viscosity approximation of HJB equations,, J. Differential Equations, 255, 4052 (2013) · Zbl 1293.45006 · doi:10.1016/j.jde.2013.07.056 |
| [5] | L. Caffarelli, Regularity theory for nonlinear integro differential equations,, Communications in Pure and Applied Mathematics, 62, 597 (2009) · Zbl 1170.45006 · doi:10.1002/cpa.20274 |
| [6] | L. Caffarelli, Drift diffusion equations with fractional diffusion and quasi geostrophic equation,, Annals of Math., 171, 1903 (2010) · Zbl 1204.35063 · doi:10.4007/annals.2010.171.1903 |
| [7] | L. Caffarelli, Regularity results for non-local equations by approximations,, Arch. Ration. Mech. Anal., 200, 59 (2011) · Zbl 1231.35284 · doi:10.1007/s00205-010-0336-4 |
| [8] | L. Caffarelli, The Evans-Krylov theorem for nonlocal fully nonlinear equations,, Ann. of Math., 174, 1163 (2011) · Zbl 1232.49043 · doi:10.4007/annals.2011.174.2.9 |
| [9] | H. C. Lara, Regularity for solutions of nonlocal parabolic equations,, Calc. var. Partial Differential Equations, 49, 139 (2014) · Zbl 1292.35068 · doi:10.1007/s00526-012-0576-2 |
| [10] | M. G. Crandall, User’s guide to viscosity solutions of second order partial differential equations,, Bull. Amer. Math. Soc. (N.S.), 27, 1 (1992) · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5 |
| [11] | J. Droniou, Fractal first-order partial differential equations,, Arch. Ration. Mech. Anal., 182, 299 (2006) · Zbl 1111.35144 · doi:10.1007/s00205-006-0429-2 |
| [12] | C. Imbert, A non-local regularization of first order Hamilton Jacobi equations,, J. Differential Equation, 211, 218 (2005) · Zbl 1073.35059 · doi:10.1016/j.jde.2004.06.001 |
| [13] | H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions,, Funkcialaj Ekvacioj, 38, 101 (1995) · Zbl 0833.35053 |
| [14] | E. R. Jakobsen, Continuous dependence estimates for viscosity solutions of integro-PDEs,, J. Differential Equations, 212, 278 (2005) · Zbl 1082.45008 · doi:10.1016/j.jde.2004.06.021 |
| [15] | A. Kiselev, Blow up and regularity for fractal Burgers equation,, Dyn. Partial Differ. Equ., 5, 211 (2008) · Zbl 1186.35020 · doi:10.4310/DPDE.2008.v5.n3.a2 |
| [16] | N. S. Landkof, <em>Osnovy Sovremennoi Teorii Ptensiala</em>,, Nauka (1966) |
| [17] | A. Sayah, Équations d’Hamilton-Jacobi du premier ordre avec termes intégro-différentiels,, Comm. Partial Differential Equations, 16, 1057 (1991) · Zbl 0742.45004 · doi:10.1080/03605309108820789 |
| [18] | L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion,, Advances in Mathematics, 226, 2020 (2011) · Zbl 1216.35165 · doi:10.1016/j.aim.2010.09.007 |
| [19] | L. Silvestre, Hölder estimates for advection fractional-diffusion equations,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11, 843 (2012) · Zbl 1263.35056 |
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