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

Abstract

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 L P estimate for for any p < ∞ in terms of and where . As a consequence, we show that a C 1,β stationary state necessarily satisfies det and is of class C 2, β in Ω. Nevertheless, singular stationary states do exist: we construct a nonsmooth C 1 solution for a particular γ in two dimensions such that det in Ω and det vanishes at precisely one point in Ω.