# Lower semicontinuity of multiple µ-quasiconvex integrals

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 9, page 105-124
- ISSN: 1292-8119

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topFragalà, Ilaria. "Lower semicontinuity of multiple µ-quasiconvex integrals." ESAIM: Control, Optimisation and Calculus of Variations 9 (2010): 105-124. <http://eudml.org/doc/90683>.

@article{Fragalà2010,

abstract = {
Lower semicontinuity results are obtained for multiple
integrals of the kind $\int _\{\mathbb\{R\}^n\}\!f(x, \!\nabla_\mu u\!)\{\rm d\} \mu$,
where μ is a given positive measure on $\mathbb\{R\}^n$, and the
vector-valued function u belongs to the Sobolev space $H
^\{1,p\}_\mu (\mathbb\{R\}^n, \mathbb\{R\}^m)$ 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.
},

author = {Fragalà, Ilaria},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Borel measures; tangent properties;
lower semicontinuity.; lower semicontinuity},

language = {eng},

month = {3},

pages = {105-124},

publisher = {EDP Sciences},

title = {Lower semicontinuity of multiple µ-quasiconvex integrals},

url = {http://eudml.org/doc/90683},

volume = {9},

year = {2010},

}

TY - JOUR

AU - Fragalà, Ilaria

TI - Lower semicontinuity of multiple µ-quasiconvex integrals

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 9

SP - 105

EP - 124

AB -
Lower semicontinuity results are obtained for multiple
integrals of the kind $\int _{\mathbb{R}^n}\!f(x, \!\nabla_\mu u\!){\rm d} \mu$,
where μ is a given positive measure on $\mathbb{R}^n$, and the
vector-valued function u belongs to the Sobolev space $H
^{1,p}_\mu (\mathbb{R}^n, \mathbb{R}^m)$ 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.

LA - eng

KW - Borel measures; tangent properties;
lower semicontinuity.; lower semicontinuity

UR - http://eudml.org/doc/90683

ER -

## References

top- E. Acerbi and N. Fusco, Semicontinuity problems in the Calculus of Variations. Arch. Rational Mech. Anal.86 (1984) 125-145.
- L. Ambrosio, Introduzione alla Teoria Geometrica della Misura e Applicazioni alle Superfici Minime, Lectures Notes. Scuola Normale Superiore, Pisa (1996).
- L. Ambrosio, On the lower-semicontinuity of quasi-convex integrals in SBV. Nonlinear Anal.23 (1994) 405-425.
- 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.
- J.M. Ball and F. Murat, W1,p-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.58 (1984) 225-253.
- G. Bouchitté and G. Buttazzo, Characterization of optimal shapes and masses through Monge-Kantorovich equation. J. Eur. Math. Soc.3 (2001) 139-168.
- G. Bouchitté, G. Buttazzo and I. Fragalà, Mean curvature of a measure and related variational problems. Ann. Scuola Norm. Sup. Pisa. Cl. Sci. IVXXV (1997) 179-196.
- G. Bouchitté, G. Buttazzo and I. Fragalà, Convergence of Sobolev spaces on varying manifolds. J. Geom. Anal.11 (2001) 399-422.
- G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a measure and applications to low dimensional structures. Calc. Var. Partial Differential Equations5 (1997) 37-54.
- 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.
- G. Bouchitté and I. Fragalà, Homogenization of elastic thin structures: A measure-fattening approach. J. Convex. Anal. (to appear).
- A. Braides, Semicontinuity, Γ-convergence and Homogenization for Multiple Integrals, Lectures Notes. SISSA, Trieste (1994).
- G. Buttazzo, Semicontinuity, Relaxation, and Integral Representation in the Calculus of Variations. Longman, Harlow, Pitman Res. Notes Math. Ser.207 (1989).
- B. Dacorogna, Direct Methods in the Calculus of Variations. Springer-Verlag, Berlin, Appl. Math. Sci. 78 (1988).
- L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. CRC Press, Ann Harbor, Stud. in Adv. Math. (1992).
- H. Federer, Geometric Measure Theory. Springer-Verlag, Berlin (1969).
- I. Fonseca and S. Müller, Quasi-convex integrands and lower semicontinuity in L1. SIAM J. Math. Anal.23 (1992) 1081-1098.
- I. Fragalà and C. Mantegazza, On some notions of tangent space to a measure. Proc. Roy. Soc. Edinburgh 129A (1999) 331-342.
- P. Hajlasz and P. Koskela, Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000).
- P. Hajlasz and P. Koskela, Sobolev meets Poincaré. C. R. Acad. Sci. Paris320 (1995) 1211-1215.
- A.D. Ioffe, On lower semicontinuity of integral functionals I and II. SIAM J. Contol Optim.15 (1997) 521-538 and 991-1000.
- J. Kristensen, Lower semicontinuity in spaces of weakly differentiable functions. J. Math. Ann.313 (1999) 653-710.
- J. Maly, Lower semicontinuity of quasiconvex integrals. Manuscripta Math.85 (1994) 419-428.
- J.P. Mandallena, Contributions à une approche générale de la régularisation variationnelle de fonctionnelles intégrales, Thèse de Doctorat. Université de Montpellier II (1999).
- P. Marcellini, Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math.51 (1985) 1-28.
- 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.
- P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, London (1995).
- C.B. Morrey, Multiple Integrals in the Calculus of Variations. Springer-Verlag, Berlin (1966).
- C. Olech, Weak lower semicontuity of integral functionals. J. Optim. Theory Appl.19 (1976) 3-16.
- T. O'Neil, A measure with a large set of tangent measures. Proc. Amer. Math. Soc.123 (1995) 2217-2221.
- D. Preiss, Geometry of measures on ${\mathbb{R}}^{n}$: Distribution, rectifiability and densities. Ann. Math.125 (1987) 573-643.
- L. Simon, Lectures on Geometric Measure Theory. Australian Nat. Univ., Proc. Centre for Math. Anal.3 (1983).
- M. Valadier, Multiapplications mesurables à valeurs convexes compactes. J. Math. Pures Appl.50 (1971) 265-297.
- V.V. Zhikov, On an extension and an application of the two-scale convergence method. Mat. Sb.191 (2000) 31-72.

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