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Some remarks about relaxation problems in the Calculus of Variations

Published online by Cambridge University Press:  14 November 2011

Fabián Flores-Bazán
Fabián Flores-Bazán
Affiliation:
Universidad de Concepción, Facultad de Ciencias FÍsicas y Matemàticas, Dpto. de IngenierÍa Matemàtica, Casilla 4009, Concepción, Chile
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We study variational problems for the functional F(u) = ∫Ω f(x, u(x), Lu(x)) dx where u∈uo + V, with Vbeing any closed linear subspace of W2.P(Ω) containing W2.p.0(Ω), Ω is a bounded open set, p > 1, L is a differential operator of second order. We determine the greatest lower semicontinuous function majorised by F for the weak topology of W2.p, for its sequential version if f satisfies no coercivity assumption, showing that in both cases the relaxed functional is expressed in terms of the function ξ↦ f**(x, u, ξ). Finally, an existence result in case f (not necessarily convex) depending only on the Laplacian, is given

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 3 , 1996 , pp. 665 - 675
Copyright
Copyright © Royal Society of Edinburgh 1996

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