für Mathematik in den Naturwissenschaften Leipzig

Motivated by relaxation in the calculus of variations, this paper addresses convex but not necessarily strictly convex minimization problems. A class of energy functionals is described for which any stress field σ in L(Ω) with div σ in W 1,p ′ (Ω) (from Euler Lagrange equations and smooth lower order terms) belongs to W 1,q loc (Ω). Applications include the scalar double-well potential, an optimal design problem, a vectorial double-well problem in a compatible case, and Hencky elastoplasticity with hardening. If the energy density depends only on the modulus of the gradient we also show regularity up to the boundary.