Differentiability properties of weak solutions of certain variational problems in the theory of perfect elastoplastic plates

Abstract

In this paper we deal with the regularity of weak solutions of some variational problems arising in the theory of perfect elastoplastic plates. All results concern differential properties of the tensor of moments, which is the solution of the dual variational problem. We show that the tensor of moments has generalized derivatives of first order which are locally square summable and prove that an open set exists where the solution is regular (the tensor of moments is Holder continuous) and some quadratic form of moments is less than a critical value. In the complement of this set the quadratic form is equal to the critical value almost everywhere. Under some additional assumptions the Lebesgue measure of the complement is zero and we have the regularity of a weak solution on the open set of full measure.