Local Invertibility in Sobolev Spaces with Applications to Nematic Elastomers and Magnetoelasticity

被引:41
作者
Barchiesi, Marco [1 ]
Henao, Duvan [2 ]
Mora-Corral, Carlos [3 ]
机构
[1] Univ Napoli Federico II, Dipartimento Matemat & Applicaz, Via Cintia, I-80126 Naples, Italy
[2] Pontificia Univ Catolica Chile, Fac Math, Vicuna Mackenna 4860, Santiago, Chile
[3] Univ Autonoma Madrid, Fac Sci, C Tomas & Valiente 7, E-28049 Madrid, Spain
关键词
LOWER SEMICONTINUITY; ENERGY MINIMIZERS; WEAK CONTINUITY; EXISTENCE; DEFORMATIONS; QUASICONVEXITY; ELASTICITY; CONVEXITY; PROPERTY;
D O I
10.1007/s00205-017-1088-1
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We define a class of deformations in , , with a positive Jacobian, that do not exhibit cavitation. We characterize that class in terms of the non-negativity of the topological degree and the equality Det = det (that the distributional determinant coincides with the pointwise determinant of the gradient). Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity. We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in , and the sequence of local inverses converges to the local inverse. We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps. We apply those results to prove the existence of minimizers in some models for nematic elastomers and magnetoelasticity.
引用
收藏
页码:743 / 816
页数:74
相关论文
共 75 条
[1]   Γ-convergence of energies for nematic elastomers in the small strain limit [J].
Agostiniani, Virginia ;
DeSimone, Antonio .
CONTINUUM MECHANICS AND THERMODYNAMICS, 2011, 23 (03) :257-274
[2]  
Ambrosio L., 2000, OX MATH M, pxviii, DOI [10.1017/S0024609301309281, 10.1093/oso/9780198502456.001.0001]
[3]  
[Anonymous], 2007, SPRINGER MONOGRAPHS
[4]  
[Anonymous], 2009, PRINCETON MATH SERIE
[5]  
[Anonymous], 1977, Herriot Watt Symposion: Nonlinear Analysis and Mechanics
[6]  
[Anonymous], 2007, Liquid Crystal Elastomers
[7]  
[Anonymous], 2014, Geometric Measure Theory. Classics in Mathematics
[8]  
[Anonymous], 1989, TRANSL MATH MONOGRAP
[9]  
[Anonymous], 2000, THEORY MATRICES
[10]  
[Anonymous], 1985, NONLINEAR FUNCTIONAL