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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcialaj Ekvacioj, 38 (1995), 101-120.

[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.

[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.

[16]

N. S. Landkof, Osnovy Sovremennoi Teorii Ptensiala, Nauka, Moscow, 1966.

[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.

[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.

[19]

L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 843-855.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

H. Ishii, On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions, Funkcialaj Ekvacioj, 38 (1995), 101-120.

[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.

[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.

[16]

N. S. Landkof, Osnovy Sovremennoi Teorii Ptensiala, Nauka, Moscow, 1966.

[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.

[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.

[19]

L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 11 (2012), 843-855.

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