Further remarks on the lower semicontinuity of polyconvex integrals.(English)Zbl 0833.49013

Summary: We study some lower semicontinuity properties of polyconvex integrals of the form $\int_\Omega f(M(\nabla u))dx,$ where $$\Omega\subset \mathbb{R}^n$$, $$u: \Omega\to \mathbb{R}^m$$, and $$M(\nabla u)$$ denotes the family of the determinants of all minors of the gradient matrix $$\nabla u$$. In particular, we study the lower semicontinuity along sequences converging strongly in $$L^1(\Omega, \mathbb{R}^m)$$ when the integrand depends only on the minors of $$\nabla u$$ up to a given order, and the lower semicontinuity along sequences converging strongly in $$L^1(\Omega, \mathbb{R}^n)$$ and bounded in $$W^{1, n-1}(\Omega, \mathbb{R}^n)$$ in the special case $$m= n$$.

MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation 49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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References:

 [1] E. Acerbi and G. Dal Maso, New lower semicontinuity results for polyconvex integrals, Calc. Var. (to appear). · Zbl 0810.49014 [2] Ball, J. M.; Murat, F., W^{1,p}-quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., Vol. 66, 439-253, (1986) · Zbl 0549.46019 [3] Buttazzo, G., Semicontinuity, relaxation and integral representation problems in the calculus of variations, Pitman Res. Notes in Math., Vol. 207, (1989), Longman Harlow · Zbl 0669.49005 [4] Dacorogna, B., Direct methods in the calculus of variation, (1989), Springer-Verlag Berlin · Zbl 0676.46035 [5] Dacorogna, B.; Marcellini, P., Semicontinuité pour des intégrands polyconvexes sans continuité des déterminants, C. R. Acad. Sci. Paris, Sér. I, Vol. 311, 393-396, (1990) · Zbl 0723.49007 [6] G. Dal Maso and C. Sbordone, Weak lower semicontinuity of policonvex integrals: a borderline case, Math. Z. (to appear). · Zbl 0822.49010 [7] Federer, H., Geometric measure theory, (1969), Springer-Verlag Berlin · Zbl 0176.00801 [8] Giaquinta, M.; Modica, G.; Souček, J., Cartesian currents, weak diffeomorphisms and nonlinear elasticity, Arch. Rational Mech. Anal., Vol. 109, 385-392, (1990) · Zbl 0712.73009 [9] Giaquinta, M.; Modica, G.; Souček, J., The Dirichlet integral for mappings between manifolds: Cartesian currents and homology, Math. Ann., Vol. 294, 325-386, (1992) · Zbl 0762.49018 [10] Giusti, E., Minimal surfaces and functions of bounded variation, (1984), Birkhäuser Boston · Zbl 0545.49018 [11] Goffman, C.; Serrin, J., Sublinear functions of measures and variational integrals, Duke Math. J., Vol. 31, 159-178, (1964) · Zbl 0123.09804 [12] Maly, J., Weak lower semicontinuity of polyconvex integrals, Proc. Roy. Soc., Vol. 123A, 681-691, (1993), Edinburgh · Zbl 0813.49017 [13] Reshetnyak, Yu. G., Weak convergence of completely additive vector functions on a set, Siberian Math. J., Vol. 101, 139-167, (1961) [14] Rockafellar, T., Convex analysis, (1970), Princeton University Press Princeton · Zbl 0193.18401 [15] Schwartz, L., Théorie des distributions, (1966), Hermann Paris [16] Simon, L. M., Lectures on geometric measure theory, Proc. of the Centre for Mathematical Analysis, Vol. 3, (1983), Australian National University Canberra · Zbl 0546.49019
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