Lower semicontinuity of multiple μ-quasiconvex integrals
ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 105-124.

Lower semicontinuity results are obtained for multiple integrals of the kind ∫ ℝ n f(x,∇ μ u)dμ, where μ is a given positive measure on ℝ n , and the vector-valued function u belongs to the Sobolev space H μ 1,p (ℝ n ,ℝ m ) associated with μ. The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to μ. More precisely, for fully general μ, a notion of quasiconvexity for f along the tangent bundle to μ, turns out to be necessary for lower semicontinuity; the sufficiency of such condition is also shown, when μ belongs to a suitable class of rectifiable measures.

DOI : 10.1051/cocv:2003002
Classification : 28A25, 49J45, 26B25
Mots clés : Borel measures, tangent properties, lower semicontinuity 
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     title = {Lower semicontinuity of multiple $\sf \mu $-quasiconvex integrals},
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Fragalà, Ilaria. Lower semicontinuity of multiple $\sf \mu $-quasiconvex integrals. ESAIM: Control, Optimisation and Calculus of Variations, Tome 9 (2003), pp. 105-124. doi : 10.1051/cocv:2003002. http://www.numdam.org/articles/10.1051/cocv:2003002/

[1] E. Acerbi and N. Fusco, Semicontinuity problems in the Calculus of Variations. Arch. Rational Mech. Anal. 86 (1984) 125-145. | MR | Zbl

[2] L. Ambrosio, Introduzione alla Teoria Geometrica della Misura e Applicazioni alle Superfici Minime, Lectures Notes. Scuola Normale Superiore, Pisa (1996). | Zbl

[3] L. Ambrosio, On the lower-semicontinuity of quasi-convex integrals in SBV. Nonlinear Anal. 23 (1994) 405-425. | MR | Zbl

[4] L. Ambrosio, G. Buttazzo and I. Fonseca, Lower-semicontinuity problems in Sobolev spaces with respect to a measure. J. Math. Pures Appl. 75 (1996) 211-224. | MR | Zbl

[5] J.M. Ball and F. Murat, W 1,p -quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984) 225-253. | MR | Zbl

[6] G. Bouchitté and G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc. 3 (2001) 139-168. | Zbl

[7] G. Bouchitté, G. Buttazzo and I. Fragalà, Mean curvature of a measure and related variational problems. Ann. Scuola Norm. Sup. Pisa. Cl. Sci. IV XXV (1997) 179-196. | Numdam | MR | Zbl

[8] G. Bouchitté, G. Buttazzo and I. Fragalà, Convergence of Sobolev spaces on varying manifolds. J. Geom. Anal. 11 (2001) 399-422. | MR | Zbl

[9] G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures. Calc. Var. Partial Differential Equations 5 (1997) 37-54. | MR | Zbl

[10] G. Bouchitté and I. Fragalà, Homogenization of thin structures by two-scale method with respect to measures. SIAM J. Math. Anal. 32 (2001) 1198-1126. | MR | Zbl

[11] G. Bouchitté and I. Fragalà, Homogenization of elastic thin structures: A measure-fattening approach. J. Convex. Anal. (to appear). | MR | Zbl

[12] A. Braides, Semicontinuity, Γ-convergence and Homogenization for Multiple Integrals, Lectures Notes. SISSA, Trieste (1994).

[13] G. Buttazzo, Semicontinuity, Relaxation, and Integral Representation in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser. 207 (1989). | MR | Zbl

[14] B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin, Appl. Math. Sci. 78 (1988). | MR | Zbl

[15] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Ann Harbor, Stud. in Adv. Math. (1992). | MR | Zbl

[16] H. Federer, Geometric Measure Theory. Springer-Verlag, Berlin (1969). | MR | Zbl

[17] I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L 1 . SIAM J. Math. Anal. 23 (1992) 1081-1098. | MR | Zbl

[18] I. Fragalà and C. Mantegazza, On some notions of tangent space to a measure. Proc. Roy. Soc. Edinburgh 129A (1999) 331-342. | MR | Zbl

[19] P. Hajlasz and P. Koskela, Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000). | MR | Zbl

[20] P. Hajlasz and P. Koskela, Sobolev meets Poincaré. C. R. Acad. Sci. Paris 320 (1995) 1211-1215. | MR | Zbl

[21] A.D. Ioffe, On lower semicontinuity of integral functionals I and II. SIAM J. Contol Optim. 15 (1997) 521-538 and 991-1000. | MR | Zbl

[22] J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. J. Math. Ann. 313 (1999) 653-710. | MR | Zbl

[23] J. Maly, Lower semicontinuity of quasiconvex integrals. Manuscripta Math. 85 (1994) 419-428. | MR | Zbl

[24] J.P. Mandallena, Contributions à une approche générale de la régularisation variationnelle de fonctionnelles intégrales, Thèse de Doctorat. Université de Montpellier II (1999).

[25] P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math. 51 (1985) 1-28. | MR | Zbl

[26] P. Marcellini and C. Sbordone, On the existence of minima of multiple integrals in the Calculus of Variations. J. Math. Pures Appl. 62 (1983) 1-9. | MR | Zbl

[27] P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, London (1995). | MR | Zbl

[28] C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer-Verlag, Berlin (1966). | MR | Zbl

[29] C. Olech, Weak lower semicontuity of integral functionals. J. Optim. Theory Appl. 19 (1976) 3-16. | MR | Zbl

[30] T. O'Neil, A measure with a large set of tangent measures. Proc. Amer. Math. Soc. 123 (1995) 2217-2221. | Zbl

[31] D. Preiss, Geometry of measures on ℝ n : Distribution, rectifiability and densities. Ann. Math. 125 (1987) 573-643. | MR | Zbl

[32] L. Simon, Lectures on Geometric Measure Theory. Australian Nat. Univ., Proc. Centre for Math. Anal. 3 (1983). | MR | Zbl

[33] M. Valadier, Multiapplications mesurables à valeurs convexes compactes. J. Math. Pures Appl. 50 (1971) 265-297. | MR | Zbl

[34] V.V. Zhikov, On an extension and an application of the two-scale convergence method. Mat. Sb. 191 (2000) 31-72. | MR | Zbl

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