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Lower semicontinuity in Sobolev spaces below the growth exponent of the integrand

Published online by Cambridge University Press:  14 November 2011

Jan Kristensen
Jan Kristensen
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, U.K
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Abstract

Let there be given a non-negative, quasiconvex function F satisfying the growth condition

for some p ∈]1, ∞[. For an open and bounded set Ω⊂ℝm, we show that if

then the variational integral

is lower semicontinuous on sequences of W1, p functions converging weakly in W1, q. In the proof, we make use of an extension operator to fix the boundary values. This idea is due to Meyers [26] and Maly [22], and the main contribution here is contained in Lemma 4.1, where a more efficient extension operator than the one in [22] (and in [14]) is used. The properties of this extension operator are in a certain sense best possible.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 127 , Issue 4 , 1997 , pp. 797 - 817
Copyright
Copyright © Royal Society of Edinburgh 1997

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References

1Acerbi, E. and Maso, G. Dal. New lower semicontinuity results for polyconvex integrals. Caic. Var. 2 (1994), 329–72.CrossRefGoogle Scholar
2Acerbi, E. and Fusco, F.. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86(1984), 125–45.CrossRefGoogle Scholar
3Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1970).Google Scholar
4Ambrosio, L. and Maso, G. Dal. On the relaation in (Ω;ℜm of quasi-convex integrals. J. Fund. Anal. 109 (1992), 76–97.CrossRefGoogle Scholar
5Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
6Ball, J. M. and Murat, F.. W 1, p quasiconvexity and variational problems for mutiple integrals. J. Fund. Anal. 58 (1984), 225–53.CrossRefGoogle Scholar
7Bergh, J. M. and Löfström, J.. Interpolation Spaces, Grundlehren der Mathematischen Wissenschaften 223 (Berlin: Springer, 1976).CrossRefGoogle Scholar
8Celada, P. and Maso, G. Dal. Further remarks on the lower semicontinuity of polyconvex integrals. Ann. lnst. H. Poincaré Anal. Non Linéaire 11 (1995), 661–91.CrossRefGoogle Scholar
9Dacorogna, B.. Direct Methods in the Calculus of Variations, Applied Mathematical Sciences 78 (Berlin: Springer, 1989).CrossRefGoogle Scholar
10Dacorogna, B. and Marcellini, P.. Semicontinuité pour des intégrandes polyconvexes sans continuité des determinants. C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 393–6.Google Scholar
11Maso, G. Dal and Sbordone, C.. Weak lower semicontinuity of polyconvex integrals: a borderline case. Math. Z. 218 (1995), 603–9.CrossRefGoogle Scholar
12Fefferman, C. and Stein, E. M.. H p spaces of several variables. Ada. Math. 129 (1972), 137–93.Google Scholar
13Flett, T. M.. On the rate of growth of mean values of holomorphic and harmonic functions. Proc. London Math. Soc. 20 (1970), 748–68.Google Scholar
14Fonseca, I. and Malý, J.. Relaxation of multiple integrals in Sobolev spaces below the growth exponent for the energy density (Preprint, Center for Nonlinear Analysis, Carnegie Mellon University, 1995).Google Scholar
15Fonseca, I. and Müller, S.. Quasiconvex integrands and lower semicontinuity in L 1. SIAM J. Math. Anal. 23(1992), 1081–98.CrossRefGoogle Scholar
16Fonseca, I. and Miiller, S.. Relaxation of quasiconvex functional in BV(Ω;ℜp for integrands f(x, u, ∇u). Arch. RAtional Mech. Anal. 123 (1993), 1–49.CrossRefGoogle Scholar
17Fusco, N. and Hutchinson, J. E.. A direct proof for lower semicontinuity of polyconvex functionals (Preprint, 1994).CrossRefGoogle Scholar
18Gangho, W.. On the weak lower semicontinuity of energies with polyconvex integrands. J. Math. Pures Appl. 73 (1994), 455–69.Google Scholar
19Kristensen, J.. Lower Semicontinuity of Variational Integrals (Ph.D. Thesis, Technical University of Denmark, 1994).Google Scholar
20Kristensen, J.. Lower Semicontinuity of Variational Integrals defined on W 1, p (to appear).Google Scholar
21Malý, J.. Weak lower semicontinuity of polyconvex integrals. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), 681–91.CrossRefGoogle Scholar
22Malý, J.. Weak lower semicontinuity of polyconvex and quasiconvex integrals (Preprint, 1993).CrossRefGoogle Scholar
23Malý, J.. Lower semicontinuity of quasiconvex integrals. Manuscripta Math. 85 (1994), 419–28.CrossRefGoogle Scholar
24Marcellini, P.. Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manuscripta Math. 51 (1985), 1–28.CrossRefGoogle Scholar
25Marcellini, P.. On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré, Anal. Non Linéaire 3 (1986), 391–409.CrossRefGoogle Scholar
26Meyers, N. G.. Quasiconvexity and the semicontinuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965), 125–49.CrossRefGoogle Scholar
27Morrey, C. B.. Quasiconvexity and the semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–53.CrossRefGoogle Scholar
28Morrey, C. B.. Multiple Integrals in the Calculus of Variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
29Rudin, W.. Real and Complex Analysis (New York: McGraw-Hill, 1986).Google Scholar
30Šverák, V.. Rank-one convexity does not imply quasiconvexity. Proc. Roy. Soc. Edinburgh Sect. A 120(1992), 185–9.CrossRefGoogle Scholar
31Taibleson, M. H.. On the theory of Lipschitz spaces of distributions of Euclidean n-space. I. Principal properties. J. Math. Mech. 13 (1964), 407–79; II. Translation invariant operators, duality, and interpolation. J. Math. Mech. 14 (1965), 821–39.Google Scholar
32Ziemer, W. P.. Weakly Differentiable Functions, Graduate Texts in Mathematics (Berlin: Springer, 1989).CrossRefGoogle Scholar