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Existence of minimizers for non-quasiconvex integrals. (English) Zbl 0837.49002

In this paper the authors give conditions for existence and non-existence of solutions of a problem of the type \[ \inf\Biggl\{F(u)= \int_\Omega f(Du(x))dx,\quad u\in u_0+ W^{1,\infty}_0(\Omega, \mathbb{R}^N)\Biggr\} \] with linear boundary data \(u_0\) and non-quasiconvex functions \(f\). Among others, their detailed analysis shows that there is more hope to solve the minimization problem in the vectorial case \((N> 1)\) than in the scalar case \((N= 1)\).
People interested in these topics can find in this paper a rich bibliography and a variety of tools and examples.
Reviewer: R.Schianchi (Roma)

MSC:

49J10 Existence theories for free problems in two or more independent variables
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
49J45 Methods involving semicontinuity and convergence; relaxation
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