Singular Perturbation as a Selection Criterion for Young‐Measure Solutions

We prove existence of Young-measure solutions of an Euler-Lagrange equation arising from a one-dimensional nonconvex variational problem in nonlinear elasticity. In particular, we consider a physically reasonable stored-energy density $W$ such that $W(x,\mu)$ goes to infinity for $\mu\searrow 0$ and...

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Bibliographic Details
Published inSIAM journal on mathematical analysis Vol. 39; no. 1; pp. 195 - 209
Main Authors Lilli, M., Healey, T. J., Kielhöfer, H.
Format Journal Article
LanguageEnglish
Published Philadelphia Society for Industrial and Applied Mathematics 01.01.2007
Subjects
Applied mathematics
Energy
Equilibrium
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Summary:We prove existence of Young-measure solutions of an Euler-Lagrange equation arising from a one-dimensional nonconvex variational problem in nonlinear elasticity. In particular, we consider a physically reasonable stored-energy density $W$ such that $W(x,\mu)$ goes to infinity for $\mu\searrow 0$ and $\mu\to\infty$. The selection criterion for the Young measure is a singular perturbation in form of an interfacial energy with capillarity coefficient . We first establish uniform a priori bounds on all solutions of the Euler-Lagrange equation, before passing to the limit for $\varepsilon\searrow 0$. Moreover, the singular perturbation allows us to characterize the support of the Young measure.
ISSN:0036-1410
1095-7154
DOI:10.1137/060656978