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

Imran H.  Biswas; Indranil  Chowdhury

doi:10.3934/cpaa.2016.15.907

Article Contents

2016, Volume 15, Issue 3: 907-927. Doi: 10.3934/cpaa.2016.15.907

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

Imran H. Biswas^1, and
Indranil Chowdhury^1,

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

Received: July 31, 2015

Revised: November 30, 2015

Published: April 2016

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Abstract

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.

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Mathematics Subject Classification: 35R11, 35F21, 45K05, 49L20, 49L25, 91A23.

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