Generalized Solutions of First Order PDEs

Subbotin Andrei I.

ISBN 10: 1461269202 / ISBN 13: 9781461269205

Editore: Springer, 2013

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Print on Demand pp. 332 49:B&W 6.14 x 9.21 in or 234 x 156 mm (Royal 8vo) Perfect Bound on White w/Gloss Lam. Codice articolo 95966327

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Titolo: Generalized Solutions of First Order PDEs

Casa editrice: Springer

Data di pubblicazione: 2013

Legatura: Brossura

Condizione: New

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Riassunto:

Hamilton-Jacobi equations and other types of partial differential equa­ tions of the first order are dealt with in many branches of mathematics, mechanics, and physics. These equations are usually nonlinear, and func­ tions vital for the considered problems are not smooth enough to satisfy these equations in the classical sense. An example of such a situation can be provided by the value function of a differential game or an optimal control problem. It is known that at the points of differentiability this function satisfies the corresponding Hamilton-Jacobi-Isaacs-Bellman equation. On the other hand, it is well known that the value function is as a rule not everywhere differentiable and therefore is not a classical global solution. Thus in this case, as in many others where first-order PDE's are used, there arises necessity to introduce a notion of generalized solution and to develop theory and methods for constructing these solutions. In the 50s-70s, problems that involve nonsmooth solutions of first­ order PDE's were considered by Bakhvalov, Evans, Fleming, Gel'fand, Godunov, Hopf, Kuznetzov, Ladyzhenskaya, Lax, Oleinik, Rozhdestven­ ski1, Samarskii, Tikhonov, and other mathematicians. Among the inves­ tigations of this period we should mention the results of S.N. Kruzhkov, which were obtained for Hamilton-Jacobi equation with convex Hamilto­ nian. A review of the investigations of this period is beyond the limits of the present book. A sufficiently complete bibliography can be found in [58, 126, 128, 141].

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