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Erschienen in:
2013 | OriginalPaper | Buchkapitel
19. Hamilton-Jacobi methods
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
In the preceding chapters, the predominant issues have been those connected with the deductive method: existence on the one hand, and the twin issues of regularity and necessary conditions on the other. We now introduce the reader to verification functions, a technique which unifies all the main inductive methods. It will be seen that this leads to a complex of ideas centered around the Hamilton-Jacobi inequality (or equation). To illustrate the method, we give an elementary proof of the celebrated logarithmic Sobolev inequality. The final section considers the following Cauchy problem for the Hamilton-Jacobi equation: The need to consider generalized solution concepts is explained, and the connection to viscosity solutions is made.
$$\mathbf{(H J)}\quad \left\{\quad \begin{aligned} u_{\,t} + H( x , u_{\, x}) &\: = \:0 ,\:\: (t, x)\,\in\, \Omega\,:=\, (0 ,\infty) \times {\mathbb{R}}^{ n}\\ u(0 , x) &\: = \:\ell(x)\,,\;\; x\in\, {\mathbb{R}}^{ n}. \end{aligned} \right. $$