Attainment and relaxation results in special classes of deformations

In this paper we deal with the attainment and relaxation issues in variational problems of mathematical theory of elasticity. We consider minimization of energy functionals in certain classes of deformations which render the problem essentially scalar. It turns out that in those cases the relaxation theorem holds for integrands that are bounded from below by a power function, with power exceeding the dimension of the space of independent variables. The bound from below can be improved in the homogeneous case. The mathematical fact behind the results is that relaxation holds at those Sobolev functions that are a.e. differentiable in the classical sense, independently of growth of Carathéodory integrands. In the homogeneous case we can also indicate a condition which is both necessary and sufficient for solvability of all boundary value minimization problems of the Dirichlet type.Received: 16 June 2000, Accepted: 12 March 2003, Published online: 6 June 2003M. A. Sychev: This research was partially supported by the grant N 00-01-00912 of the Russian Foundation for Basic Research