# American Institute of Mathematical Sciences

May  2016, 15(3): 907-927. doi: 10.3934/cpaa.2016.15.907

## On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian

 1 Centre for Applicable Mathematics, Tata Institute of Fundamental Research, P.O. Box 6503, GKVK Post Office, Bangalore 560065, India, India

Received  August 2015 Revised  December 2015 Published  February 2016

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.
Citation: Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907
##### References:
 [1] G. Barles, E. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations,, \emph{Indiana Univ. Math. J.}, 57 (2008), 213. doi: 10.1512/iumj.2008.57.3315. Google Scholar [2] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited,, \emph{Ann. Inst. H. Poincare Anal. Non Linaire}, 25 (2008), 567. doi: 10.1016/j.anihpc.2007.02.007. Google Scholar [3] I. H. Biswas, On zero-sum stochastic differential games with jump-diffusion driven state: a viscosity solution framework,, \emph{SIAM J. Control Optim.}, 50 (2012), 1823. doi: 10.1137/080720504. Google Scholar [4] I. H. Biswas, Regularization by $\frac{1}{2}$-Laplacian and vanishing viscosity approximation of HJB equations,, \emph{J. Differential Equations}, 255 (2013), 4052. doi: 10.1016/j.jde.2013.07.056. Google Scholar [5] L. Caffarelli and L. Silvestre, Regularity theory for nonlinear integro differential equations,, \emph{Communications in Pure and Applied Mathematics}, 62 (2009), 597. doi: 10.1002/cpa.20274. Google Scholar [6] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and quasi geostrophic equation,, \emph{Annals of Math.}, 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903. Google Scholar [7] L. Caffarelli and L. Silvestre, Regularity results for non-local equations by approximations,, \emph{Arch. Ration. Mech. Anal.}, 200 (2011), 59. doi: 10.1007/s00205-010-0336-4. Google Scholar [8] L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations,, \emph{Ann. of Math.}, 174 (2011), 1163. doi: 10.4007/annals.2011.174.2.9. Google Scholar [9] H. C. Lara and G. Davila, Regularity for solutions of nonlocal parabolic equations,, \emph{Calc. var. Partial Differential Equations}, 49 (2014), 139. doi: 10.1007/s00526-012-0576-2. Google Scholar [10] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [11] J. Droniou and C. Imbert, Fractal first-order partial differential equations,, \emph{Arch. Ration. Mech. Anal.}, 182 (2006), 299. doi: 10.1007/s00205-006-0429-2. Google Scholar [12] C. Imbert, A non-local regularization of first order Hamilton Jacobi equations,, \emph{J. Differential Equation}, 211 (2005), 218. doi: 10.1016/j.jde.2004.06.001. Google Scholar [13] H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions,, \emph{Funkcialaj Ekvacioj}, 38 (1995), 101. Google Scholar [14] E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of integro-PDEs,, \emph{J. Differential Equations}, 212 (2005), 278. doi: 10.1016/j.jde.2004.06.021. Google Scholar [15] A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation,, \emph{Dyn. Partial Differ. Equ.}, 5 (2008), 211. doi: 10.4310/DPDE.2008.v5.n3.a2. Google Scholar [16] N. S. Landkof, Osnovy Sovremennoi Teorii Ptensiala,, Nauka, (1966). Google Scholar [17] A. Sayah, Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels,, \emph{Comm. Partial Differential Equations}, 16 (1991), 1057. doi: 10.1080/03605309108820789. Google Scholar [18] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion,, \emph{Advances in Mathematics}, 226 (2011), 2020. doi: 10.1016/j.aim.2010.09.007. Google Scholar [19] L. Silvestre, Hölder estimates for advection fractional-diffusion equations,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 11 (2012), 843. Google Scholar

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##### References:
 [1] G. Barles, E. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations,, \emph{Indiana Univ. Math. J.}, 57 (2008), 213. doi: 10.1512/iumj.2008.57.3315. Google Scholar [2] G. Barles and C. Imbert, Second-order elliptic integro-differential equations: Viscosity solutions' theory revisited,, \emph{Ann. Inst. H. Poincare Anal. Non Linaire}, 25 (2008), 567. doi: 10.1016/j.anihpc.2007.02.007. Google Scholar [3] I. H. Biswas, On zero-sum stochastic differential games with jump-diffusion driven state: a viscosity solution framework,, \emph{SIAM J. Control Optim.}, 50 (2012), 1823. doi: 10.1137/080720504. Google Scholar [4] I. H. Biswas, Regularization by $\frac{1}{2}$-Laplacian and vanishing viscosity approximation of HJB equations,, \emph{J. Differential Equations}, 255 (2013), 4052. doi: 10.1016/j.jde.2013.07.056. Google Scholar [5] L. Caffarelli and L. Silvestre, Regularity theory for nonlinear integro differential equations,, \emph{Communications in Pure and Applied Mathematics}, 62 (2009), 597. doi: 10.1002/cpa.20274. Google Scholar [6] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and quasi geostrophic equation,, \emph{Annals of Math.}, 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903. Google Scholar [7] L. Caffarelli and L. Silvestre, Regularity results for non-local equations by approximations,, \emph{Arch. Ration. Mech. Anal.}, 200 (2011), 59. doi: 10.1007/s00205-010-0336-4. Google Scholar [8] L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations,, \emph{Ann. of Math.}, 174 (2011), 1163. doi: 10.4007/annals.2011.174.2.9. Google Scholar [9] H. C. Lara and G. Davila, Regularity for solutions of nonlocal parabolic equations,, \emph{Calc. var. Partial Differential Equations}, 49 (2014), 139. doi: 10.1007/s00526-012-0576-2. Google Scholar [10] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations,, \emph{Bull. Amer. Math. Soc. (N.S.)}, 27 (1992), 1. doi: 10.1090/S0273-0979-1992-00266-5. Google Scholar [11] J. Droniou and C. Imbert, Fractal first-order partial differential equations,, \emph{Arch. Ration. Mech. Anal.}, 182 (2006), 299. doi: 10.1007/s00205-006-0429-2. Google Scholar [12] C. Imbert, A non-local regularization of first order Hamilton Jacobi equations,, \emph{J. Differential Equation}, 211 (2005), 218. doi: 10.1016/j.jde.2004.06.001. Google Scholar [13] H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions,, \emph{Funkcialaj Ekvacioj}, 38 (1995), 101. Google Scholar [14] E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of integro-PDEs,, \emph{J. Differential Equations}, 212 (2005), 278. doi: 10.1016/j.jde.2004.06.021. Google Scholar [15] A. Kiselev, F. Nazarov and R. Shterenberg, Blow up and regularity for fractal Burgers equation,, \emph{Dyn. Partial Differ. Equ.}, 5 (2008), 211. doi: 10.4310/DPDE.2008.v5.n3.a2. Google Scholar [16] N. S. Landkof, Osnovy Sovremennoi Teorii Ptensiala,, Nauka, (1966). Google Scholar [17] A. Sayah, Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels,, \emph{Comm. Partial Differential Equations}, 16 (1991), 1057. doi: 10.1080/03605309108820789. Google Scholar [18] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion,, \emph{Advances in Mathematics}, 226 (2011), 2020. doi: 10.1016/j.aim.2010.09.007. Google Scholar [19] L. Silvestre, Hölder estimates for advection fractional-diffusion equations,, \emph{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 11 (2012), 843. Google Scholar
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