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On the definition and the lower semicontinuity of certain quasiconvex integrals. (English) Zbl 0609.49009
The author studies the lower semicontinuity in the weak topology of \(H^{1,P}(\Omega;R^ N)\) of the functional \[ I(u)=\int_{\Omega}f(x,Du(x))dx, \] where f(x,z) is a continuous function in \(\Omega \times R^{nN}\), quasiconvex with respect on z, that satisfies for some \(p\leq q\) the following growth condition \(c_ 1| z|^ p\leq f(x,z)\leq c_ 2(1+| z|)^ q)\). Since I(u) is well defined if \(u\in H^{1,q}(\Omega;R^ N)\), the author has first to extend the integral to functions \(u\in H^{1,P}(\Omega,R^ N)\) and after that he studies the semicontinuity in order to obtain existence of minima.
Reviewer: R.Schianchi

49J45 Methods involving semicontinuity and convergence; relaxation
26B25 Convexity of real functions of several variables, generalizations
35J50 Variational methods for elliptic systems
Full Text: DOI Numdam EuDML
[1] Acerbi, E.; Fusco, N., Semicontinuity problems in the calculus of variations, Arch. Rat. Mech. Anal., t. 86, 125-145, (1984) · Zbl 0565.49010
[2] Antman, S. S., The influence of elasticity on analysis: modern developments, Bull. Amer. Math. Soc., t. 9, 267-291, (1983) · Zbl 0533.73001
[3] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., t. 63, 337-403, (1977) · Zbl 0368.73040
[4] Ball, J. M., Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond., t. 306, 557-611, (1982) · Zbl 0513.73020
[5] Ball, J. M.; Murat, F., W^{1,p}-quasiconvexity and variational problems for multiple integrals, J. Functional Analysis, t. 58, 225-253, (1984) · Zbl 0549.46019
[6] Dacorogna, B., Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Math., t. 922, (1982), Springer-Verlag Berlin · Zbl 0676.46035
[7] Dal Maso, G., Integral representation on BV(ω) of γ-limits of variational integrals, Manuscripta Math., t. 30, 387-416, (1980) · Zbl 0435.49016
[8] E. De Giorgi, Teoremi di semicontinuità nel calcolo delle variazioni, Istituto Nazionale di Alta Matematica, Roma, 1968-1969.
[9] De Giorgi, E., Sulla convergenza di alcune successioni di integrali del tipo dell’area, Rendiconti Mat., t. 8, 277-294, (1975) · Zbl 0316.35036
[10] C. L. Evans, Quasiconvexity and partial regularity in the calculus of variations, preprint. · Zbl 0623.49008
[11] Ferro, F., Functionals defined on functions of bounded variation in ℝ^{n} and the Lebesgue area, SIAM J. Control Optimization, t. 16, 778-789, (1978) · Zbl 0382.46017
[12] Giaquinta, M.; Modica, G.; Soucek, J., Functionals with linear growth in the calculus of variations I, Commentationes Math. Univ. Carolinae, t. 20, 143-172, (1979) · Zbl 0409.49006
[13] M. Giaquinta, G. Modica, Partial regularity of minimizers of quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse non linéaire, to appear.
[14] Giusti, E., Non-parametric minimal surfaces with discontinuous and thin obstacles, Arch. Rat. Mech. Anal., t. 49, 41-56, (1972) · Zbl 0257.49015
[15] Knops, R. J.; Stuart, C. A., Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity, Arch. Rat. Mech. Anal., t. 86, 233-249, (1984) · Zbl 0589.73017
[16] R. V. Kohn, G. Strang, Optimal design and relaxation of variational problems, preprint.
[17] Lebesgue, H., Intégrale, longueur, aire, Ann. Mat. Pura Appl., t. 7, 231-359, (1902) · JFM 33.0307.02
[18] Marcellini, P., Quasiconvex quadratic forms in two dimensions, Appl. math. Optimization, t. 11, 183-189, (1984) · Zbl 0567.49007
[19] Marcellini, P., Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, Manuscripta Math., t. 51, 1-28, (1985) · Zbl 0573.49010
[20] Marcellini, P.; Sbordone, C., On the existence of minima of multiple integrals of the calculus of variations, J. Math. Pures Appl., t. 62, 1-9, (1983) · Zbl 0516.49011
[21] Meyers, N., Quasiconvexity and lower semicontinuity of multiple integrals of any order, Trans. Amer. Math. Soc., t. 119, 125-149, (1965) · Zbl 0166.38501
[22] Miranda, M., Un teorema di esistenza e unicità per il problema dell’area minima in n variabili, Ann. Scuola Norm. Sup. Pisa, t. 19, 233-249, (1965) · Zbl 0137.08201
[23] Morrey, C. B., Multiple integrals in the calculus of variations, (1966), Springer-Verlag Berlin · Zbl 0142.38701
[24] Serrin, J., On the definition and properties of certain variational integrals, Trans. Amer. Math. Soc., t. 101, 139-167, (1961) · Zbl 0102.04601
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