Local Stress Regularity in Scalar Nonconvex Variational Problems

In light of applications to relaxed problems 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 $\sigma$ in $L^q(\Omega)$ with $\operatorname{{\rm div}}\sigma$ in $W^{1,p'}(\Omega)$ belongs to $ W^{1,q}_{loc}(\Omega)$. The condition on $\operatorname{{\rm div}}\sigma$ holds, for example, for solutions of the Euler--Lagrange equations involving additional lower-order terms. 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.