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Maximal smoothness of solutions to certain Euler–Lagrange equations from nonlinear elasticity

Published online by Cambridge University Press:  14 November 2011

Patricia Bauman ,
Daniel Phillips  and
Nicholas C. Owen
Patricia Bauman
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN. 47907, U.S.A.
Daniel Phillips
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN. 47907, U.S.A.
Nicholas C. Owen
Affiliation:
Department of Applied and Computational Mathematics, The University of Sheffield, Sheffield, S10 2TN, England
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Synopsis

We investigate the maximal smoothness of stationary states for the multiple integral =

Such variational problems are motivated by the study of nonlinear elasticity. Assuming certain structure conditions for γ and given a stationary state , we derive an a priori LP estimate for for any p < ∞ in terms of and where . As a consequence, we show that a C1,β stationary state necessarily satisfies det and is of class C2, β in Ω. Nevertheless, singular stationary states do exist: we construct a nonsmooth C1 solution for a particular γ in two dimensions such that det in Ω and det vanishes at precisely one point in Ω.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 119 , Issue 3-4 , 1991 , pp. 241 - 263
Copyright
Copyright © Royal Society of Edinburgh 1991

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