Diffeomorphic Approximation of planar Elastic Deformations

Language
en
Document Type
Doctoral Thesis
Issue Date
2017-09-18
Issue Year
2017
Authors
Radici, Emanuela
Editor
Abstract

This Thesis is devoted to the problem of � finding regular approximations of planar homeomorphisms with respect to several suitable distances having important applications in nonlinear elasticity. By regular approximations we intend either di� ffeomorphisms or piecewise a� ne homeomorphisms. Both the results would be interesting in di� fferent contexts. However, in the two dimensional setting the two things are equivalent and, hence, we can focus directly on the approximation via piecewise a� ne homeomorphisms that is, in general, a bit easier task. Before analyzing the approximation problem, we discuss the possible applications in nonlinear elasticity and the functional spaces where it is useful to look for the approximations. The fi� rst result of the Thesis is a generalization to the case p = n of the existence of optimal admissible deformations for the pure displacement boundary value problem that was proved by Ball in W1 ;p for p > n, where n indicates the dimension of the reference con� figuration. For this purpose, we recall some known properties of the mappings of � nite distortion and of the bi-Sobolev homeomorphisms. Then we focus on the approximation of homeomorphisms in the Sobolev, bi-Sobolev and BV spaces. We present a uni� ed argument showing that any planar W1 ;p homeomorphism can be approximated in W1 ;p and uniformly by piecewise a� ne homeomorphisms, where p varies in [1 ;1). To do this we extend the tools available for the case p = 1 to every p > 1. After that, we deal with the problem of approximating also the inverse mappings. We recall the already known results in the bi-W1 ;1 and bi-W1 ;1 spaces and we perform a fi� ner analysis for our main result: the sharp approximation of planar homeomorphisms of bounded variation. The principal di� fficulty in this setting is that the singular part of the distributional derivative of the homeomorphism may not be zero, thus we cannot allow for less precise approximations even in a very small portion of the domain. For this reason, we need to use a very sophisticated Extension Theorem, that is the main novelty of this Thesis. At last, we brie y discuss a possible approach to deal with homeomorphisms in the space bi-W1 ;p when p > 1. Despite the promising results for p = 1 ;1, the problem of approximating strongly in bi-W1 ;p is still open in its generality.

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