Partial Regularity for Almost Minimizers of Quasi-Convex Integrals

We consider almost minimizers of variational integrals whose integrands are quasi-convex. Under suitable growth conditions on the integrand and on the function determining the almost minimality, we establish almost everywhere regularity for almost minimizers and obtain results on the regularity of the gradient away from the singular set. We give examples of problems from the calculus of variations whose solutions can be viewed as such almost minimizers.