Γ-convergence of equi-coercive nonlinear energies defined on vector-valued functions, with non-uniformly bounded coefficients. (March 2017)

• Record Type: Journal Article
• Authors: Briane, M.; Casado-Díaz, J.; Luna-Laynez, M.; Pallares-Martín, A.
• Is Part Of: Nonlinear analysis. Volume 151(2017)
• Page Start: 187
• Page End: 207
• Publication Date: 2017-03
• Subjects: 35B27 -- 74B20; Γ-convergence -- Nonlinear elliptic systems -- Non-uniformly bounded coefficients -- Hyperelasticity; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.11.009
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: The present paper deals with the asymptotic behavior of equi-coercive sequences { ℱ n } of nonlinear functionals defined over vector-valued functions in W 0 1, p ( Ω ) M, where p > 1, M ≥ 1, and Ω is a bounded open set of R N, N ≥ 2 . The strongly local energy density F n ( ⋅, D u ) of the functional ℱ n satisfies a Lipschitz condition with respect to the second variable, which is controlled by a positive sequence { a n } which is only bounded in some suitable space L r ( Ω ) . We prove that the sequence { ℱ n } Γ -converges for the strong topology of L p ( Ω ) M to a functional ℱ which has a strongly local density F ( ⋅, D u ) for sufficiently regular functions u . This compactness result extends former results on the topic, which are based either on maximum principle arguments in the nonlinear scalar case, or adapted div–curl lemmas in the linear case. Here, the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional ℱ n . The relevance of the conditions which are imposed to the energy density F n ( ⋅, D u ), is illustrated by several examples including some classical hyperelastic energies.