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

Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Proceedings of the Royal Society of Edinburgh: Section A Mathematics / Volume 119 / Issue 3-4 / January 1991, pp 241-263
Copyright © Royal Society of Edinburgh 1991
DOI: http://dx.doi.org/10.1017/S0308210500014815 (About DOI), Published online: 14 November 2011

Table of Contents - 1991 - Volume 119, Issue 3-4

Research Article

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

Article author query
bauman p [Google Scholar]
phillips d [Google Scholar]
owen nc [Google Scholar]

Patricia Bauman^a1*, Daniel Phillips^a1^† and Nicholas C. Owen^a2

^a1Department of Mathematics, Purdue University, West Lafayette, IN. 47907, U.S.A.

^a2Department of Applied and Computational Mathematics, The University of Sheffield, Sheffield, S10 2TN, England

Synopsis

We investigate the maximal smoothness of stationary states for the multiple integral S0308210500014815_inline1 = S0308210500014815_inline2

Such variational problems are motivated by the study of nonlinear elasticity. Assuming certain structure conditions for γ and given a stationary state S0308210500014815_inline3 , we derive an a priori L^P estimate for S0308210500014815_inline4 for any p < ∞ in terms of S0308210500014815_inline5 and S0308210500014815_inline6 where S0308210500014815_inline7 . As a consequence, we show that a C^1,β stationary state necessarily satisfies det S0308210500014815_inline8 and is of class C^{2, 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 S0308210500014815_inline9 in Ω and det S0308210500014815_inline10 vanishes at precisely one point in Ω.