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.

[1] D.G. Aronson, Bounds for Fundamental Solution of a Parabolic Equation. Bull. Amer. Math. Soc. 73 (1968) 890-896. | MR | Zbl

[2] A. Bensoussan and J. Frehse, Regularity of Solutions of Systems of Partial Differential Equations and Applications. Springer Verlag (to be published).

[3] A. Bensoussan and J. Frehse, Nonlinear elliptic systems in stochastic game theory. J. Reine Angew. Math. 350 (1984) 23-67. | MR | Zbl

[4] A. Bensoussan and J. Frehse, $C^{\alpha }$-Regularity Results for Quasi-Linear Parabolic Systems. Comment. Math. Univ. Carolin. 31 (1990) 453-474. | MR | Zbl

[5] A. Bensoussan and J. Frehse, Ergodic Bellman systems for stochastic games, in Differential equations, dynamical systems, and control science. Dekker, New York (1994) 411-421. | MR | Zbl

[6] A. Bensoussan and J. Frehse, Ergodic Bellman systems for stochastic games in arbitrary dimension. Proc. Roy. Soc. London Ser. A 449 (1935) 65-77. | MR | Zbl

[7] A. Bensoussan and J. Frehse, Stochastic games for $N$ players. J. Optim. Theory Appl. 105 (2000) 543-565. Special Issue in honor of Professor David G. Luenberger. | MR | Zbl

[8] A. Bensoussan and J.-L. Lions, Impulse control and quasivariational inequalities. Gauthier-Villars (1984). Translated from the French by J.M. Cole. | MR

[9] S. Campanato, Equazioni paraboliche del secondo ordine e spazi $L^{2,\theta }(\Omega ,\delta )$. Ann. Mat. Pura Appl. (4) 73 (1966) 55-102. | MR | Zbl

[10] G. Da Prato, Spazi $L^{(p,\theta )}(\Omega ,\delta )$ e loro proprietà. Ann. Mat. Pura Appl. (4) 69 (1965) 383-392. | MR | Zbl

[11] J. Frehse, Remarks on diagonal elliptic systems, in Partial differential equations and calculus of variations. Springer, Berlin (1988) 198-210. | MR | Zbl

[12] J. Frehse, Bellman Systems of Stochastic Differential Games with three Players in Optimal Control and Partial Differential Equations, edited by J.L. Menaldi, E. Rofman and A. Sulem. IOS Press (2001). | Zbl

[13] S. Hildebrandt and K.-O. Widman, Some regularity results for quasilinear elliptic systems of second order. Math. Z. 142 (1975) 67-86. | MR | Zbl

[14] J. Leray and J.-L. Lions, Quelques résultats de Višik sur les problèmes elliptiques nonlinéaires par les méthodes de Minty-Browder. Bull. Soc. Math. France 93 (1965) 97-107. | Numdam | Zbl

[15] O.A. Ladyženskaja, V.A. Solonnikov and N.N. Ural'Ceva, Linear and quasilinear equations of parabolic type. American Mathematical Society, Providence, R.I. (1967). | MR | Zbl

[16] M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems. Manuscripta Math. 35 (1981) 125-145. | MR | Zbl

[17] M. Wiegner, Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme. Math. Z. 147 (1976) 21-28. | MR | Zbl