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MiS Preprint

Local stress regularity in scalar non-convex variational problems

Carsten Carstensen and Stefan Müller


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 $\sigma$ in $L^q(\Omega)$ with div $\sigma$ in $ W^{1,p'}(\Omega)$ (from Euler Lagrange equations and smooth lower order terms) belongs to $ W^{1,q}_{loc}$ $(\Omega)$. 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.

MSC Codes:
49J45, 35B65, 35J60
non-convex minimization, regularization, relaxed problem, stress regularity

Related publications

2002 Repository Open Access
Carsten Carstensen and Stefan Müller

Local stress regularity in scalar nonconvex variational problems

In: SIAM journal on mathematical analysis, 34 (2002) 2, 495-509 (electronic)