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On lower semicontinuity of multiple integrals. (English) Zbl 0893.49009

The paper presents a short proof of the lower semicontinuity of the integral functional \[ \int_\Omega f(x,u,\nabla u) dx,\qquad u:\Omega\subset{\mathbb{R}}^n\to{\mathbb{R}}^k \] in the weak topology of \(W^{1,p}\) under suitable \(p\)-growth conditions, and quasi convexity of \(z\mapsto f(x,s,z)\). The proof is based on the analysis of the Young measures generated by the sequences \((u_h,\nabla u_h)\) and on the dual inequality \[ \int g(z) d\nu(z)\geq g\biggl(\int z d\nu(z)\biggr) \] between quasi convex functions \(g\) with \(p\) growth and homogeneous gradient Young measure \(\nu\) generated by a sequence bounded in \(W^{1,p}\).
Reviewer: L.Ambrosio (Pavia)

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q20 Variational problems in a geometric measure-theoretic setting
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