## Lower semicontinuity of quasiconvex integrals.(English)Zbl 0862.49017

Summary: New lower semicontinuity results for quasiconvex integrals are established. In particular, under certain structure conditions and growth $$0\leq F(\xi)\leq C(1+|\xi|^q)$$ the functional $$\int_{\Omega}F(\nabla u)dx$$ is proved to be lower semicontinuous on $$W^{1,q}$$ with respect to the weak convergence in $$W^{1,p}$$, $$p\geq q-1$$.

### MSC:

 49J45 Methods involving semicontinuity and convergence; relaxation

### Keywords:

semicontinuity; quasiconvex integrals; weak convergence
Full Text:

### References:

 [1] Acerbi, E. and Dal Maso, G.: New lower semicontinuity results for polyconvex integrals case. Preprint SISSA, Trieste52/M (1993), · Zbl 0810.49014 [2] Ball, J. M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal.63 (1977), 337–403. · Zbl 0368.73040 [3] Ball, J. M. and Murat, F.:W 1,p quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984), 225–253 · Zbl 0549.46019 [4] Carbone, L. and De Arcangelis, R.: Further results on {$$\Gamma$$}-convergence and lower semicontinuity of integral functionals depending on vector-valued functions. Richerche Mat.39 (1990), 99–129. · Zbl 0735.49008 [5] Celada, P. and Dal Maso, G.: Further remarks on the lower semicontinuity of polyconvex integrals. Ann. Inst. Henri Poincaré (to appear). · Zbl 0833.49013 [6] Dacorogna, B.: Direct Methods in the Calculus of Variations, Applied Math. Sciences 78, Springer-Verlag, 1989. · Zbl 0703.49001 [7] Dacorogna, B. and Marcellini, P.: Semicontinuité pour des intégrandes polyconvexes sans continuité des determinants. C. R. Acad. Sci. Paris Sér. I Math.311, 6 (1990), 393–396. · Zbl 0723.49007 [8] Dal Maso, G. and Sbordone, C.: Weak lower semicontinuity of polyconvex integrals: a borderline case. Preprint SISSA, Trieste,45/M (1993), · Zbl 0822.49010 [9] Federer, H.: Geometric measure theory, Springer-Verlag, Grundlehren, 1969. · Zbl 0176.00801 [10] Fonseca, I. and Malý, J.: Relaxation of Multiple Integrals in Sobolev Spaces below the growth exponent for the energy density. In preparation. · Zbl 0868.49011 [11] Fonseca, I. and Malý, J.: Relaxation of Multiple Integrals under Constraints. In preparation. · Zbl 0868.49011 [12] Fonseca, I. and Marcellini, P.: Relaxation of multiple integrals in subcritical Sobolev spaces. Preprint 1994. · Zbl 0915.49011 [13] Fonseca, I. and Müller, S.: Quasiconvex integrands and lower semicontunuity inL 1. SIAM J. Math. Anal. (to appear). [14] Fusco, N. and Hutchinson, J. E.: A direct proof for lower semicontinuity of polyconvex functionals. Preprint 1994. · Zbl 0874.49015 [15] Gangbo, W.: On the weak lower semicontinuity of energies with polyconvex integrands. Preprint C.M.U., Pittsburgh 1992. · Zbl 0829.49011 [16] Giaquinta, M., Modica, G. and Souček, J.: Cartesian currents and variational problems for mappings into spheres. Ann. Sc. Norm. Sup. Pisa16 (1989), 393–485. · Zbl 0713.49014 [17] Malý, J.: Weak lower semicontinuity of polyconvex integrals. Proc. Royal Soc. Edinburgh123A, 681–691. (1993), · Zbl 0813.49017 [18] Malý, J. Weak lower semicontinuity of polyconvex and quasiconvex integrals. Vortragsreihe 1993 (to appear), Bonn. · Zbl 0813.49017 [19] Marcellini, P.: Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manus. Math.51 (1985), 1–28. · Zbl 0573.49010 [20] Marcellini, P.: On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré, Analyse non Linéaire3 (1986), 391–409. · Zbl 0609.49009 [21] Meyers, N. G.: Quasiconvexity and the semicontinuity of multiple variational integrals of any order. Trans. Amer. Math. Soc.119 (1965), 125–149. · Zbl 0166.38501 [22] Morrey, C. B.: Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math.,2 (1952), 25–53. · Zbl 0046.10803 [23] –: Multiple integrals in the Calculus of Variations, Springer, Berlin, 1966. · Zbl 0142.38701 [24] Müller, S.: Weak continuity of determinants and nonlinear elasticity. C. R. Acad. Sci. Paris307 (1988), 501–506. · Zbl 0679.34051 [25] Müller, S., Tang, Q. and Yan, B.S.: On a new class of elastic deformations not allowing for cavitation. Preprint 255, University of Bonn 1992. · Zbl 0863.49002 [26] Šverák, V.: Regularity properties of deformations with finite energy. Arch. Rat. Mech. Anal.100 (1988), 105–127. · Zbl 0659.73038
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