Attainment and relaxation results in special classes of deformations (Q1879315)

The paper deals with attainment and relaxation issues in variational problems of mathematical theory of elasticity. Minimization problems are considered for energy functionals in certain classes of deformations which allow a reduction to essentially scalar cases. Relaxation results are established for integrands bounded from below by a power function, with exponent exceeding the independent variables space dimension. Bounds from below are improved in the homogeneous case. The mathematical fact exploited to achieve such results is that the relaxation result holds for those Sobolev functions that are a.e. differentiable in the classical sense, independently on the growth of the integrand. In the homogeneous case, a condition is indicated which is both necessary and sufficient for the solvability of all boundary value minimization problems of Dirichlet type.