Semicontinuity problems in the calculus of variations. (English) Zbl 0537.49002

We are concerned with integral functionals of the form \(F(\Omega,u)=\int_{\Omega}f(x,u,Du)dx,\) where \(\Omega\) is a bounded open set of \({\mathbb{R}}^ n\) and \(u\in H^{1,p}(\Omega)\) for some \(p\geq 1\). Throughout the paper we assume that \(f:{\mathbb{R}}^ n\times {\mathbb{R}}\times {\mathbb{R}}^ n\to {\mathbb{R}}\) satisfies the following conditions: (a) \(0\leq f(x,s,\xi)\leq g(x,| s|,| \xi |),\) with g increasing with respect to \(| s|\) and \(| \xi |\), and locally summable in x. (b) f(x,s,\(\xi)\) is measurable with respect to x, upper semicontinuous with respect to \(\xi\), and continuous with respect to s uniformly as \(\xi\) varies on each bounded set of \({\mathbb{R}}^ n\). Two meaningful cases in which (b) is satisfied are the following: (i) \(f=f(x,\xi)\) is measurable in x and upper semicontinuous in \(\xi\) ; (ii) f is a Carathéodory function, i.e. measurable in x and continuous in \((x,\xi)\). We prove that the convexity of f with respect to \(\xi\) is a necessary condition for the s.l.s. (sequential lower semicontinuity) of F and we consider the following problem: if f is not convex with respect to \(\xi\), which is the greatest functional (s.l.s. in the weak topology o \(H^{1,p}(\Omega))\) which is less than or equal to \(F(\Omega\),\(\cdot)?\) We list some cases in which it is possible to characterize this functional as the integral \(\int_{\Omega}f^{**}(x,u,Du)dx,\) where \(f^{**}(x,s,\xi)\) is the greatest function (convex in \(\xi)\) which is less than or equal to f. Some unsolved problems remain when \(f^{**}\) is not a Carathéodory function. We apply these results to the relaxation of some variational problems. Relaxation means that, starting from a minimum problem for F which lacks a solution, a second minimum problem (the relaxed or generalized problem) is formulated involving a certain integral with the same infimum and whose optimal solutions are the limit points of the sequences minimizing the first problem. We study the relaxation for the Dirichlet and Neumann problems, for the obstacle problem and the relaxation for the convex set of functions with a prescribed bound on the Lipschitz constant.


49J45 Methods involving semicontinuity and convergence; relaxation
49M20 Numerical methods of relaxation type
49Q20 Variational problems in a geometric measure-theoretic setting
90C25 Convex programming
49L99 Hamilton-Jacobi theories
54C08 Weak and generalized continuity
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[1] Serrin, J., On the definition and properties of certain variational integrals, Trans. Am. math. Soc., 101, 139-167 (1961) · Zbl 0102.04601
[2] de Giorgi, E., Teoremi di semicontinuità nel calcolo delle variazioni (1968-1969), Istit. Naz. Alta Mat: Istit. Naz. Alta Mat Roma
[3] Berkowitz, L. D., Lower semicontinuity of integral functionals, Trans. Am. math. Soc., 192, 51-57 (1974) · Zbl 0294.49001
[4] Cesari, L., Lower semicontinuity and lower closure theorems without seminormality condition, Annali Mat. pura appl., 98, 381-397 (1974) · Zbl 0281.49006
[5] Ioffe, A. D., On lower semicontinuity of integral functionals I, SIAM J. Cont. Optimization, 15, 521-538 (1977) · Zbl 0361.46037
[6] Olech, C., A characterization of \(L^1\)-weak lower semicontinuity of integral functionals, Bull. Acad. pol. Sci. Sér. Sci. Math. Astronom. Phys., 25, 135-142 (1977) · Zbl 0395.46026
[7] Ball, J. M., On the calculus of variations and sequentially weakly continuous maps. Ordinary and P.D.E., (Lecture Notes in Math., 564 (1976), Springer: Springer Berlin), 13-25, Dundee · Zbl 0348.49004
[8] Tonelli, L.; Zanichelli, Fondamenti di Calcolo Delle Variazioni, Vol. I (1921)
[9] Caccioppoli, R.; Scorza Dragoni, G., Necessità della condizione di Weierstrass per la semicontinuità di un integrale doppio sopra una data superficie, Memorie Acc. d’Italia, 9, 251-268 (1938) · JFM 64.0514.02
[10] McShane, B. J., On the necessary condition of Weierstrass in the multiple integral problem of the calculus of variations, Ann. Math., 32, 578-590 (1931) · Zbl 0003.06002
[11] Morrey, C. B., Multiple Integrals in the Calculus of Variations (1966), Springer: Springer Berlin · Zbl 0142.38701
[12] Ekeland, I.; Temam, R., Analyse convexe et problèmes variationels (1974), Dunod & Gauthier: Dunod & Gauthier Villars · Zbl 0281.49001
[13] Marcellini, P.; Sbordone, C., Relaxation of non convex variational problems, Rend. Acc. Naz. Lincei, 63, 341-344 (1977) · Zbl 0408.49017
[14] Goofman, G.; Serrin, J., Sublinear functions of measures and variational integrals, Duke math. J., 31, 159-178 (1964) · Zbl 0123.09804
[15] Ferro, F., Functionals defined on functions of bounded variation in \(\textbf{R}^n\) and the Lebesgue area, SIAM J. Cont. Optimization, 16, 778-789 (1978) · Zbl 0382.46017
[18] Marcellini, P.; Pitagora, Proc. Inter. Meeting “Recent Meth. Non Linear-Analysis”, Rome 1978. Proc. Inter. Meeting “Recent Meth. Non Linear-Analysis”, Rome 1978, Some problems of semicontinuity and of Γ-convergence for integrals of the calculus of variations, 205-221 (1979), Bologna · Zbl 0405.49021
[19] Rockafellar, R. T., Integral functionals, normal integrands and measurable selections, (Nonlinear operators and the calc. of variant.. Nonlinear operators and the calc. of variant., Lecture Notes in Math., 543 (1975), Springer: Springer Berlin), 157-207, Bruxelles · Zbl 0374.49001
[20] Halmos, P. R., Measure Theory (1974), Springer: Springer Berlin · Zbl 0117.10502
[21] Rockafellar, R. T., Convex Analysis (1970), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 0229.90020
[22] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer Berlin · Zbl 0691.35001
[23] Krasnosel’skii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations (1964), Pergamon Press: Pergamon Press Oxford · Zbl 0111.30303
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