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Erschienen in:
2013 | OriginalPaper | Buchkapitel
20. Multiple integrals
verfasst von : Francis Clarke
Erschienen in: Functional Analysis, Calculus of Variations and Optimal Control
Verlag: Springer London
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Abstract
The problem under consideration is the minimization of the functional over a class X of real-valued functions u defined on Ω, where Ω denotes a nonempty open bounded subset of \({\mathbb{R}}^{ n}\). The values of the admissible functions are prescribed on the boundary of Ω : where φ is given. Of course, the reader will recognize this as an extension to multiple integrals of the basic problem in the calculus of variations; this chapter serves as an introduction to that topic. As in the single integral case, it is perhaps the choice of underlying space X that makes the greatest difference in the ensuing theory.
$$\quad J(u)\:\, =\:\: \int_\Omega F\big( x , u(x),\,D u(x)\big)\, dx , $$
$$u(x)\: = \:\varphi(x)~\; \forall \,x\: \in \:\Gamma\, :=\:\, \partial\Omega\,, $$