Parabolic Equations with Quadratic Growth in R^n
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AbstractWe study here quasi-linear parabolic equations with quadratic growth in Rn . These parabolic equations are at the core of the theory of PDE; see Friedman (Partial differential equations of parabolic type. Prentice-Hall, Englewood Cliffs, 1964) [6], Ladyzhenskaya et al. (Translations of Mathematical Monographs. AMS, 1968) [4] for details. However, for the applications to physics and mechanics, one deals mostly with boundary value problems. The boundary is often taken to be bounded and the solution is bounded. This brings an important simplification. On the other hand, stochastic control theory leads mostly to problems in Rn . Moreover, the functions are unbounded and the Hamiltonian may have quadratic growth. There may be conflicts which prevent solutions to exist. In stochastic control theory, a very important development deals with BSDE (Backward Stochastic Differential Equations). There is a huge interaction with parabolic PDE in Rn . This is why, although we do not deal with BSDE in this paper, we use many ideas from Briand and Hu (Probab Theory Relat Fields 141(3–4):543–567, 2008) [1], Da Lio and Ley (SIAM J Control Optim 45(1):74–106, 2006) [2], Karoui et al. (Backward stochastic differential equations and applications, Princeton BSDE Lecture Notes, 2009) [3], Kobylanski (Ann Probab 28(2):558–602, 2000) [5]. Our presentation provided here is slightly innovative.
Acceptance Date20/07/2018
All Author(s) ListAlain Bensoussan, Jens Frehse, Shige Peng, Sheung Chi Phillip Yam
All Editor(s) ListB. N. Chetverushkin, W. Fitzgibbon, Y.A. Kuznetsov, P. Neittaanmäki, J. Periaux, O. Pironneau
EditionFirst
Book titleContributions to Partial Differential Equations and Applications
Series TitleComputational Methods in Applied Sciences
Year2018
Month7
Volume Number47
PublisherSpringer Verlag
Place of PublicationUSA
Pages91 - 110
ISBN978-3-319-78324-6
eISBN978-3-319-78325-3
LanguagesEnglish-United States
KeywordsParabolic Equations, Quadratic Growth

Last updated on 2020-15-08 at 02:54