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Cartesian currents and variational problems for mappings into spheres. (English) Zbl 0713.49014
This paper studies variational problems with constraints for vector valued mappings. It is strongly related to the work of M. Giaquinta, G. Modica and J. Soucek, Arch. Ration. Mech. Anal. 106, No.2, 97-159 (1989; Zbl 0677.73014).
Section 2 contains a presentation of several classes of cartesian currents and the problem of the convergence of determinants on the basis of the above mentioned results. In particular, relationships among boundaries, traces and weak convergence are discussed. Section 3 is devoted to the notion of degree: degree for cartesian currents, degree and weak diffeomorphisms, etc.
In Section 4 the authors define the polyconvex extension of a general integrand as, roughly, the largest polyconvex integrand which is below the given one. They also introduce the parametric integrand associated with such an extension and compute these extensions for some specific cases. Furthermore, they discuss the problem of the existence of energy minimizing maps with prescribed degree from an n-dimensional Riemannian manifold into \(S^ n\). They prove the existence of a minimizer for regular functionals.
In Section 5 the authors treat several problems in which one looks for minimizers of the Dirichlet integral and of the more general functional of liquid crystals, among mappings from a domain of \({\mathbb{R}}^ 3\) into \(S^ 2\) satisfying suitable “boundary conditions”. The existence of a minimizer is shown. Section 6 is devoted to some extensions, nonlinear hyperelasticity and fractures.
Reviewer: R.Euler

MSC:
49J40 Variational inequalities
74B20 Nonlinear elasticity
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
49Q20 Variational problems in a geometric measure-theoretic setting
Citations:
Zbl 0677.73014
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References:
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