Abstract
We prove the local Hölder continuity of strong local minimizers of the stored energy functionalE(u)=∫Ωλ|∇u|2+h(det∇u)dx\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$E(u)=\\int_{\\Omega}\\lambda |\\nabla u|^{2}+h({\\rm det} \\nabla u)\\,{\\rm d}x$$\\end{document}subject to a condition of ‘positive twist’. The latter turns out to be equivalent to requiring that u maps circles to suitably star-shaped sets. The convex function h(s) grows logarithmically as {sto 0+}, linearly as {s to +infty}, and satisfies {h(s)=+infty} if {s leqq 0}. These properties encode a constitutive condition which ensures that material does not interpenetrate during a deformation and is one of the principal obstacles to proving the regularity of local or global minimizers. The main innovation is to prove that if a local minimizer has positive twist a.e. on a ball then an Euler-Lagrange type inequality holds and a Caccioppoli inequality can be derived from it. The claimed Hölder continuity then follows by adapting some well-known elliptic regularity theory. We also demonstrate the regularizing effect that the term {int_{Omega} h({rm det} nabla u),{rm d}x} can have by analysing the regularity of local minimizers in the class of ‘shear maps’. In this setting a more easily verifiable condition than that of positive twist is imposed, with the result that local minimizers are Hölder continuous.
Highlights
In this paper we consider the question of the regularity of local minimizers of functionals representing the stored energy of two-dimensional elastic bodies
Ball notes in [3] that this is one of a number of outstanding open problems in the field, and while we believe that the results presented here are a positive contribution towards improving the regularity of elastic energy minimizers, we do not claim to be able to prove that the latter are smooth, which in the sense of [3, Section
The constitutive condition (1.1) models the physical reality that compressing material to zero volume ought to incur an infinite energetic cost. That such a condition could be captured and embedded in an existence theory using polyconvex functions was first realized by Ball [1], whose well-known work has since given rise to a rich literature on the topic
Summary
In this paper we consider the question of the regularity of local minimizers of functionals representing the stored energy of two-dimensional elastic bodies. Foss [8] uses a blow-up technique to establish a partial regularity result for minimizers u, say, of functionals in which (1.1) is operational For his result to hold, u is required to satisfy an equiintegrability condition, referred to as (REP), which is phrased in terms of an excess quantity. With this in force, u is shown to be C1,α on an open set of full measure in Ω It is not known whether property (REP) holds for minimizers of stored-energy functionals. 3 to a particular class of maps—the so-called shear maps These allow us to treat a much wider class of stored-energy functions in the sense that the h(det ∇u) term of the stored-energy function W is required to obey much weaker growth conditions: see hypotheses (H0)–(H4) of Sect.
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