Relaxation of multiple integrals below the growth exponent. (English) Zbl 0868.49011

Summary: The integral representation of the relaxed energies \[ \begin{split}{\mathcal F}^{q,p}(u,\Omega):=\inf_{\{u_n\}} \Biggl\{\liminf_{n\to\infty} \int_\Omega F(x,u_n,\nabla u_n) dx: u_n\in W^{1,q}(\Omega,\mathbb{R}^d),\\ u_n\rightharpoonup u\text{ weakly in }W^{1,p}(\Omega,\mathbb{R}^d)\Biggr\},\end{split} \]
\[ \begin{split} {\mathcal F}^{q,p}_{\text{loc}}(u,\Omega):= \inf_{\{u_n\}} \Biggl\{\liminf_{n\to\infty} \int_\Omega F(x,u_n,\nabla u_n) dx: u_n\in W^{1,q}_{\text{loc}}(\Omega,\mathbb{R}^d),\\ u_n\rightharpoonup u\text{ weakly in }W^{1,p}(\Omega,\mathbb{R}^d)\Biggr\}\end{split} \] of a functional \[ E: u\mapsto \int_\Omega F(x,u,\nabla u) dx,\quad u\in W^{1,q}(\Omega,\mathbb{R}^d), \] where \(0\leq F(x,\zeta,\xi)\leq C(1+|\zeta|^r+ |\xi|^q)\) and \(\max\{1,r{N-1\over N+r},q{N-1\over N}\}< p\leq q\), is studied. In particular, \(W^{1,p}\)-sequential weak lower semicontinuity of \(E(\cdot)\) is obtained in the case where \(F=F(\xi)\) is a quasiconvex function and \(p>q(N-1)/N\).


49J45 Methods involving semicontinuity and convergence; relaxation
49Q10 Optimization of shapes other than minimal surfaces
Full Text: DOI Numdam EuDML


[1] Acerbi, E.; Dal Maso, G., New lower semicontinuity results for polyconvex integrals case, Preprint SISSA, Trieste, Vol. 52/M (1993) · Zbl 0810.49014
[2] Acerbi, E.; Fusco, F., Semicontinuity problems in the Calculus of variations, Arch. Rat. Mech. Anal., Vol. 86, 125-145 (1984) · Zbl 0565.49010
[3] Ball, J. M., Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rat. Mech. Anal., Vol. 63, 337-403 (1977) · Zbl 0368.73040
[4] Ball, J. M.; Murat, F., \(W^{1,p}\) quasiconvexity and variational problems for multiple integrals, J. Funct. Anal., Vol. 58, 225-253 (1984) · Zbl 0549.46019
[5] Carbone, L.; De Arcangelis, R., Further results on Γ-convergence and lower semicontinuity of integral functionals depending on vector-valued functions, Richerche Mat., Vol. 39, 99-129 (1990) · Zbl 0735.49008
[7] Dacorogna, B., Direct Methods in the Calculus of Variations, (Applied Math. Sciences, 78 (1989), Springer-Verlag) · Zbl 0547.49003
[8] Dacorogna, B.; Marcellini, P., Semicontinuité pour des intégrandes polyconvexes sans continuité des determinants, C. R. Acad. Sci. Paris Sér. I Math., Vol. 311, 6, 393-396 (1990) · Zbl 0723.49007
[9] Dal Maso, G.; Sbordone, C., Weak lower semicontinuity of polyconvex integrals: a borderline case, Preprint SISSA, Trieste, Vol. 45 (1993) · Zbl 0822.49010
[12] Fusco, N.; Hutchinson, J. E., A direct proof for lower semicontinuity of polyconvex functionals (1994), Preprint
[13] Gangbo, W., On the weak lower semicontinuity of energies with polyconvex integrands, J. Math. Pures et Appl., Vol. 73, 5 (1994) · Zbl 0829.49011
[15] Malý, J., Weak lower semicontinuity of polyconvex integrals, (Proc. Royal Soc. Edinburgh, Vol. 123A (1993)), 681-691 · Zbl 0813.49017
[16] Malý, J., Weak lower semicontinuity of polyconvex and quasiconvex integrals, Vortragsreihe (1993), Bonn (to appear) · Zbl 0813.49017
[17] Malý, J., Lower semicontinuity of quasiconvex integrals, Manuscripta Math., Vol. 85, 419-428 (1994) · Zbl 0862.49017
[18] Marcellini, P., Approximation of quasiconvex functions and lower semicontinuity of multiple integrals quasiconvex integrals, Manuscripta Math., Vol. 51, 1-28 (1985) · Zbl 0573.49010
[19] Marcellini, P., On the definition and the lower semicontinuity of certain quasiconvex integrals, Ann. Inst. H. Poincaré, Analyse non Linéaire, Vol. 3, 391-409 (1986) · Zbl 0609.49009
[20] Meyers, N. G., Quasiconvexity and the semicontinuity of multiple variational integrals of any order, Trans. Amer. Math. Soc., Vol. 119, 125-149 (1965) · Zbl 0166.38501
[21] Morrey, C. B., Quasiconvexity and the semicontinuity of multiple integrals, Pacific J. Math., Vol. 2, 25-53 (1952) · Zbl 0046.10803
[22] Morrey, C. B., Multiple integrals in the Calculus of Variations (1966), Springer: Springer Berlin · Zbl 0142.38701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.