# Smooth Solutions of systems of quasilinear parabolic equations

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 8, page 169-193
- ISSN: 1292-8119

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topBensoussan, Alain, and Frehse, Jens. "Smooth Solutions of systems of quasilinear parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 169-193. <http://eudml.org/doc/90644>.

@article{Bensoussan2010,

abstract = {
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.
},

author = {Bensoussan, Alain, Frehse, Jens},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Parabolic equations; quasilinear; game theory; regularity; Stochastic optimal
control; smallness condition; specific structure; maximum principle; Green function;
Hamiltonian.; quadratic nonlinearity; stochastic optimal control},

language = {eng},

month = {3},

pages = {169-193},

publisher = {EDP Sciences},

title = {Smooth Solutions of systems of quasilinear parabolic equations},

url = {http://eudml.org/doc/90644},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Bensoussan, Alain

AU - Frehse, Jens

TI - Smooth Solutions of systems of quasilinear parabolic equations

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 8

SP - 169

EP - 193

AB -
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.

LA - eng

KW - Parabolic equations; quasilinear; game theory; regularity; Stochastic optimal
control; smallness condition; specific structure; maximum principle; Green function;
Hamiltonian.; quadratic nonlinearity; stochastic optimal control

UR - http://eudml.org/doc/90644

ER -

## References

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- M. Wiegner, Ein optimaler Regularitätssatz für schwache Lösungen gewisser elliptischer Systeme. Math. Z.147 (1976) 21-28. Copyright EDP Sciences, SMAI 2002

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