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

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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

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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

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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

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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

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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

show all references

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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