Abstract
A lower semicontinuity and relaxation result with respect to weak-* convergence of measures is derived for functionals of the form
where admissible sequences {μ n } are such that \({\{{\mathcal{A}}\mu_{n}\}}\) converges to zero strongly in \({W^{-1 q}_{\rm loc}(\Omega)}\) and \({\mathcal {A}}\) is a partial differential operator with constant rank. The integrand f has linear growth and L ∞-bounds from below are not assumed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams R.A.: Sobolev spaces, pure and applied mathematics, 65. In: A Series of Monographs and Textbooks. Academic Press, New York–San Francisco–London (1975)
Alberti G: Rank-one property for derivatives of functions with bounded variation. Proc. Royal Soc. Edinburgh Sect. A 123, 237–274 (1993)
Alibert JJ, Bouchitté G: Non-uniform integrability and generalized Young measures. J. Convex Anal. 4, 129–147 (1997)
Ambrosio L, Dal Maso G: On the Relaxation in \({BV(\Omega; \mathbb{R}^m)}\) of quasi convex integrals. J. Funct. Anal. 109, 76–97 (1992)
Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000)
Braides A, Fonseca I, Leoni G: \({{\mathcal A}}\)-quasiconvexity: relaxation and homogenization. ESAIM Control Optim. Calc. Var. 5, 539–577 (2000)
Dacorogna B.: Weak continuity and weak lower semicontinuity of non-linear functionals. In: Lecture Notes in Mathematics. Springer, New York (1982)
Fonseca I., Leoni G.: Modern methods in the calculus of variations: L p Spaces. In: Springer Monographs in Mathematics. Springer, New York (2007)
Fonseca I, Muller S: \({{\mathcal A}}\)-quasiconvexity, lower semicontinuity and Young measures. SIAM J. Math. Anal. 30(6), 1355–1390 (1999)
Fonseca I, Leoni G, Muller S: \({{\mathcal A}}\)-quasiconvexity: weak-star convergence and the gap. Ann. Inst. H. Poincaré Anal. Non Linéaire 21(2), 209–236 (2004)
Fonseca I, Muller S: Relaxation of quasiconvex integrals in \({BV(\Omega, \mathbb{R}^p)}\) for integrands \({f(x,u,\nabla u)}\). Arch. Ration. Mech. Anal. 123, 1–49 (1993)
Evans L.C., Gariepy R.F.: Measure theory and fine properties of functions, studies in advanced mathematics. CRC Press, London (1992)
Kristensen J, Rindler F: Relaxation of signed integrals in BV, Calc. Var. Partial Differ. Equ. 37(1–2), 29–62 (2010)
Muller S: On quasiconvex functions which are homogeneous of degree-1. Indiana Univ. Math. J 41, 295–301 (1992)
Murat F: Compacité par compensation: condition n’ecessaire et suffisante de continuité faible sous une hypothese de rang constante. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 8(1), 69–102 (1981)
Preiss D: Geometry of measures in \({\mathbb{R}b^N}\): distributions, rectifiability and densities. Ann. Math. 125, 537–643 (1987)
Rindler R: Lower semicontinuity for functions of bounded deformation via rigidity and young measures. Arch. Ration. Mech. Anal. 202, 63–113 (2011)
Rindler R.: Lower semicontinuity and Young measures in BV without Alberti’s rank-one theorem. In: Advances in Calculus of Variations. University of Oxford, Oxford (2012)
Tartar, L.: Compensated compactness and applications to partial differential equations, nonlinear analysis and mechanics: Heriot–Watt Symposium. In: Knops, R. (ed.) Pitman Research Notes in Mathematics, vol. IV, pp. 136–212 (1979)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by L. Ambrosio.
Rights and permissions
About this article
Cite this article
Baía, M., Chermisi, M., Matias, J. et al. Lower semicontinuity and relaxation of signed functionals with linear growth in the context of \({\mathcal A}\)-quasiconvexity. Calc. Var. 47, 465–498 (2013). https://doi.org/10.1007/s00526-012-0524-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-012-0524-1