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On various semiconvex hulls in the calculus of variations. (English) Zbl 0896.49005
Let \(M^{N\times n}\) be the set of the \(N\times n\) matrices and \(K\subseteq M^{N\times n}\). In the paper some relationships among some semiconvex hulls of \(K\) are studied. The closed lamination convex hull \(L_c(K)\) of \(K\), the rank-one convex hull \(R(K) \) of \(K\), the quasiconvex hull \(Q(K)\) of \(K\), the polyconvex hull \(P(K)\) of \(K\), and the convex hull \(C(K)\) of \(K\) are recalled and some of their properties are reported. Then it is proved that if \(K\) is compact and \(L_c(K)\neq C(K)\), then also \(Q(K)\) is different from \(C(K)\). In the case \(N=n=2\) it is also proved that if \(K\) is compact and \(L_c(K)\neq C(K)\) then \(P(K)\) is not convex. An estimate of the non convexity of \(Q(K)\) if \(R(K)\) is not convex is obtained by using quasiconvex envelopes of some distance functions from \(K\).

49J45 Methods involving semicontinuity and convergence; relaxation
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