Smooth solutions of systems of quasilinear parabolic equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8  (2002), p. 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.

DOI : https://doi.org/10.1051/cocv:2002059
Classification:  35XX,  49XX
Keywords: parabolic equations, quasilinear, game theory, regularity, stochastic optimal control, smallness condition, specific structure, maximum principle, Green function, hamiltonian
@article{COCV_2002__8__169_0,
     author = {Bensoussan, Alain and Frehse, Jens},
     title = {Smooth solutions of systems of quasilinear parabolic equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     pages = {169-193},
     doi = {10.1051/cocv:2002059},
     zbl = {1078.35022},
     mrnumber = {1932949},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2002__8__169_0}
}
Bensoussan, Alain; Frehse, Jens. Smooth solutions of systems of quasilinear parabolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) , pp. 169-193. doi : 10.1051/cocv:2002059. http://www.numdam.org/item/COCV_2002__8__169_0/

[1] D.G. Aronson, Bounds for Fundamental Solution of a Parabolic Equation. Bull. Amer. Math. Soc. 73 (1968) 890-896. | MR 217444 | Zbl 0153.42002

[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 743532 | Zbl 0531.93052

[4] A. Bensoussan and J. Frehse, C α -Regularity Results for Quasi-Linear Parabolic Systems. Comment. Math. Univ. Carolin. 31 (1990) 453-474. | MR 1078480 | Zbl 0726.35032

[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 1243215 | Zbl 0830.90142

[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 1328140 | Zbl 0833.90141

[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 1783877 | Zbl 0977.91006

[8] A. Bensoussan and J.-L. Lions, Impulse control and quasivariational inequalities. Gauthier-Villars (1984). Translated from the French by J.M. Cole. | MR 756234

[9] S. Campanato, Equazioni paraboliche del secondo ordine e spazi L 2,θ (Ω,δ). Ann. Mat. Pura Appl. (4) 73 (1966) 55-102. | MR 213737 | Zbl 0144.14101

[10] G. Da Prato, Spazi L (p,θ) (Ω,δ) e loro proprietà. Ann. Mat. Pura Appl. (4) 69 (1965) 383-392. | MR 192330 | Zbl 0145.16207

[11] J. Frehse, Remarks on diagonal elliptic systems, in Partial differential equations and calculus of variations. Springer, Berlin (1988) 198-210. | MR 976236 | Zbl 0669.35040

[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 1054.91511

[13] S. Hildebrandt and K.-O. Widman, Some regularity results for quasilinear elliptic systems of second order. Math. Z. 142 (1975) 67-86. | MR 377273 | Zbl 0317.35040

[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 0132.10502

[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 241822 | Zbl 0174.15403

[16] M. Struwe, On the Hölder continuity of bounded weak solutions of quasilinear parabolic systems. Manuscripta Math. 35 (1981) 125-145. | MR 627929 | Zbl 0519.35007

[17] M. Wiegner, Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme. Math. Z. 147 (1976) 21-28. | MR 407430 | Zbl 0316.35039