Smooth solutions of systems of quasilinear parabolic equations

Bensoussan, Alain; Frehse, Jens

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 169-193

We consider in this article diagonal parabolic systems arising in the context of stochastic differential games. We address the issue of finding smooth solutions of the system. Such a regularity result is extremely important to derive an optimal feedback proving the existence of a Nash point of a certain class of stochastic differential games. Unlike in the case of scalar equation, smoothness of solutions is not achieved in general. A special structure of the nonlinear hamiltonian seems to be the adequate one to achieve the regularity property. A key step in the theory is to prove the existence of Hölder solution.

Classification : 35XX, 49XX
Keywords: parabolic equations, quasilinear, game theory, regularity, stochastic optimal control, smallness condition, specific structure, maximum principle, Green function, hamiltonian

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