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$$W^{1,p}$$-quasiconvexity and variational problems for multiple integrals. (English) Zbl 0549.46019
Variational problems for the multiple integral $$I_{\Omega}(u)=\int_{\Omega}g(\nabla u(x))dx,$$ where $$\Omega\subset {\mathbb{R}}^ m$$ and $$u:\Omega\to {\mathbb{R}}^ n$$ are studied. A new condition on g, called $$W^{1,p}$$-quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of C. B. Morrey, it being shown in particular to be necessary for sequential weak lower semicontinuity of $$I_{\Omega}$$ in $$W^{1,p}(\Omega;{\mathbb{R}}^ n)$$ and for the existence of minimizers for certain related integrals. Counterexamples are given concerning the weak continuity properties of Jacobians in $$W^{1,p}(\Omega;{\mathbb{R}}^ n), p\leq n=m$$. An existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses.

##### MSC:
 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 49J27 Existence theories for problems in abstract spaces 74B20 Nonlinear elasticity
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##### References:
 [1] {\scE. Acerbi, G. Buttazzo and N. Fusco}, Semicontinuity in L∞ for polyconvex integrals, Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Natur., in press. · Zbl 0481.49013 [2] Acerbi, E; Buttazzo, G; Fusco, N, Semicontinuity and relaxation for integrals depending on vector valued functions, J. math. pures et appl., 62, 371-387, (1983) · Zbl 0481.49013 [3] {\scE. Acerbi and N. Fusco}, Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal., in press. · Zbl 0565.49010 [4] Ball, J.M, Convexity conditions and existence theorems in nonlinear elasticity, Arch. rat. mech. anal., 63, 337-403, (1977) · Zbl 0368.73040 [5] Ball, J.M, Constitutive inequalities and existence theorems in nonlinear elastostatics, (), 187-241 · Zbl 0377.73043 [6] Ball, J.M, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Philos. trans. roy. soc. London A, 306, 557-611, (1982) · Zbl 0513.73020 [7] Ball, J.M; Currie, J.C; Olver, P.J, Null Lagrangians, weak continuity, and variational problems of arbitrary order, J. funct. anal., 41, 135-174, (1981) · Zbl 0459.35020 [8] Dacorogna, B, Quasiconvexity and relaxation of non convex problems in the calculus of variations, J. funct. anal., 46, 102-118, (1982) · Zbl 0547.49003 [9] Dacorogna, B, Weak continuity and weak lower semi-continuity of nonlinear functionals, () · Zbl 0676.46035 [10] Fusco, N, Remarks on the relaxation of integrals of the calculus of variations, (), 401-408 [11] Kohn, R.V; Strang, G, Explicit relaxation of a variational problem in optimal design, Bull. amer. math. soc., 9, 211-214, (1983) · Zbl 0527.49002 [12] Meyers, N.G, Quasi-convexity and lower semicontinuity of multiple variational integrals of any order, Trans. amer. math. soc., 119, 125-149, (1965) · Zbl 0166.38501 [13] Morrey, C.B, Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. math., 2, 25-53, (1952) · Zbl 0046.10803 [14] Morrey, C.B, Multiple integrals in the calculus of variations, (1966), Springer Berlin · Zbl 0142.38701 [15] Murat, F, Compacité par compensation II, (), 245-256 · Zbl 0427.35008 [16] Reshetnyak, Y.G, On the stability of conformal mappings in multidimensional spaces, Siberian math. J., 8, 69-85, (1967) · Zbl 0172.37801 [17] Reshetnyak, Y.G, General theorems on semicontinuity and on convergence with a functional, Siberian math. J., 8, 801-816, (1967) · Zbl 0179.20902 [18] Reshetnyak, Y.G, Stability theorems for mappings with bounded excursion, Siberian math. J., 9, 499-512, (1968) · Zbl 0176.03503 [19] Saks, S, Theory of the integral, (1937), Hafner New York [20] Tartar, L, Compensated compactness and partial differential equations, (), 136-212 · Zbl 0437.35004 [21] {\scP. Marcellini}, Approximation of quasiconvex functions and lower semicontinuity of multiple integrals, to appear. · Zbl 0573.49010
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