# 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, Indiana Univ. Math. J., 57 (2008), 213-246. 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, Ann. Inst. H. Poincare Anal. Non Linaire, 25 (2008), 567-585. 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, SIAM J. Control Optim., 50 (2012), 1823-1858. doi: 10.1137/080720504.  Google Scholar [4] I. H. Biswas, Regularization by $\frac{1}{2}$-Laplacian and vanishing viscosity approximation of HJB equations, J. Differential Equations, 255 (2013), 4052-4080. doi: 10.1016/j.jde.2013.07.056.  Google Scholar [5] L. Caffarelli and L. Silvestre, Regularity theory for nonlinear integro differential equations, Communications in Pure and Applied Mathematics, 62 (2009), 597-638. doi: 10.1002/cpa.20274.  Google Scholar [6] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and quasi geostrophic equation, Annals of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar [7] L. Caffarelli and L. Silvestre, Regularity results for non-local equations by approximations, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4.  Google Scholar [8] L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math., 174 (2011), 1163-1187. doi: 10.4007/annals.2011.174.2.9.  Google Scholar [9] H. C. Lara and G. Davila, Regularity for solutions of nonlocal parabolic equations, Calc. var. Partial Differential Equations, 49 (2014), 139-172. 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, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [11] J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331. doi: 10.1007/s00205-006-0429-2.  Google Scholar [12] C. Imbert, A non-local regularization of first order Hamilton Jacobi equations, J. Differential Equation, 211 (2005), 218-246. 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, Funkcialaj Ekvacioj, 38 (1995), 101-120.  Google Scholar [14] E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of integro-PDEs, J. Differential Equations, 212 (2005), 278-318. 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, Dyn. Partial Differ. Equ., 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2.  Google Scholar [16] N. S. Landkof, Osnovy Sovremennoi Teorii Ptensiala, Nauka, Moscow, 1966.  Google Scholar [17] A. Sayah, Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels, Comm. Partial Differential Equations, 16 (1991), 1057-1093. doi: 10.1080/03605309108820789.  Google Scholar [18] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion, Advances in Mathematics, 226 (2011), 2020-2039. doi: 10.1016/j.aim.2010.09.007.  Google Scholar [19] L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 843-855.  Google Scholar

show all references

##### References:
 [1] G. Barles, E. Chasseigne and C. Imbert, On the Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J., 57 (2008), 213-246. 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, Ann. Inst. H. Poincare Anal. Non Linaire, 25 (2008), 567-585. 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, SIAM J. Control Optim., 50 (2012), 1823-1858. doi: 10.1137/080720504.  Google Scholar [4] I. H. Biswas, Regularization by $\frac{1}{2}$-Laplacian and vanishing viscosity approximation of HJB equations, J. Differential Equations, 255 (2013), 4052-4080. doi: 10.1016/j.jde.2013.07.056.  Google Scholar [5] L. Caffarelli and L. Silvestre, Regularity theory for nonlinear integro differential equations, Communications in Pure and Applied Mathematics, 62 (2009), 597-638. doi: 10.1002/cpa.20274.  Google Scholar [6] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and quasi geostrophic equation, Annals of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar [7] L. Caffarelli and L. Silvestre, Regularity results for non-local equations by approximations, Arch. Ration. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4.  Google Scholar [8] L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math., 174 (2011), 1163-1187. doi: 10.4007/annals.2011.174.2.9.  Google Scholar [9] H. C. Lara and G. Davila, Regularity for solutions of nonlocal parabolic equations, Calc. var. Partial Differential Equations, 49 (2014), 139-172. 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, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1-67. doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [11] J. Droniou and C. Imbert, Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182 (2006), 299-331. doi: 10.1007/s00205-006-0429-2.  Google Scholar [12] C. Imbert, A non-local regularization of first order Hamilton Jacobi equations, J. Differential Equation, 211 (2005), 218-246. 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, Funkcialaj Ekvacioj, 38 (1995), 101-120.  Google Scholar [14] E. R. Jakobsen and K. H. Karlsen, Continuous dependence estimates for viscosity solutions of integro-PDEs, J. Differential Equations, 212 (2005), 278-318. 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, Dyn. Partial Differ. Equ., 5 (2008), 211-240. doi: 10.4310/DPDE.2008.v5.n3.a2.  Google Scholar [16] N. S. Landkof, Osnovy Sovremennoi Teorii Ptensiala, Nauka, Moscow, 1966.  Google Scholar [17] A. Sayah, Équations d'Hamilton-Jacobi du premier ordre avec termes intégro-différentiels, Comm. Partial Differential Equations, 16 (1991), 1057-1093. doi: 10.1080/03605309108820789.  Google Scholar [18] L. Silvestre, On the differentiability of the solution to the Hamilton-Jacobi equation with critical fractional diffusion, Advances in Mathematics, 226 (2011), 2020-2039. doi: 10.1016/j.aim.2010.09.007.  Google Scholar [19] L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 843-855.  Google Scholar
 [1] Monica Motta, Caterina Sartori. Uniqueness of solutions for second order Bellman-Isaacs equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 739-765. doi: 10.3934/dcds.2008.20.739 [2] Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023 [3] Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065 [4] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [5] Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907 [6] Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020044 [7] Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053 [8] Roberto Garrappa, Eleonora Messina, Antonia Vecchio. Effect of perturbation in the numerical solution of fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2679-2694. doi: 10.3934/dcdsb.2017188 [9] Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061 [10] Liping Luo, Zhenguo Luo, Yunhui Zeng. New results for oscillation of fractional partial differential equations with damping term. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020336 [11] Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $q$-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440 [12] Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261 [13] N. V. Krylov. Uniqueness for Lp-viscosity solutions for uniformly parabolic Isaacs equations with measurable lower order terms. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2495-2516. doi: 10.3934/cpaa.2018119 [14] Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021030 [15] Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277 [16] Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020100 [17] Yin Yang, Sujuan Kang, Vasiliy I. Vasil'ev. The Jacobi spectral collocation method for fractional integro-differential equations with non-smooth solutions. Electronic Research Archive, 2020, 28 (3) : 1161-1189. doi: 10.3934/era.2020064 [18] Ankit Kumar, Kamal Jeet, Ramesh Kumar Vats. Controllability of Hilfer fractional integro-differential equations of Sobolev-type with a nonlocal condition in a Banach space. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021016 [19] Alain Bensoussan, Jens Frehse, Jens Vogelgesang. Systems of Bellman equations to stochastic differential games with non-compact coupling. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1375-1389. doi: 10.3934/dcds.2010.27.1375 [20] Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161

2019 Impact Factor: 1.105