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A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems. (English) Zbl 0809.49017

For the following type of variational problems
\(\min \int_ \Omega W(\nabla y(x))dx\),
\(\Omega\subset \mathbb{R}^ n\) open and bounded, \(y: \Omega\to \mathbb{R}\), \(y(x)\Bigl|_{\partial\Omega}= F.x\), \(F\in \mathbb{R}^ n\),
\(W\) continuous and bounded,
those integrands \(W\) are determined for which the minimum is not attained. The result generalizes a large number of known facts and counterexamples. The behavior of the minimizing sequences in the case of nonattainment is described. Two kind of proofs (depending on the use of Young measures) are presented.
Reviewer: E.Iwanow (Wien)

MSC:

49K10 Optimality conditions for free problems in two or more independent variables
49J10 Existence theories for free problems in two or more independent variables
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