a1 Department of Mathematics, Purdue University, West Lafayette, IN. 47907, U.S.A.
a2 Department of Applied and Computational Mathematics, The University of Sheffield, Sheffield, S10 2TN, England
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.
(Received January 08 1991)
* Partially supported by NSF Grant No. DMS-8912473
† Partially supported by NSF Grant No. DMS-8601515.