Convex functional and the stratification of the singular set of their stationary points

Abstract

We prove partial regularity of stationary solutions and minimizers $u$ from a set $\Omega \subset {\mathbb{R}}^n$ to a Riemannian manifold $N$, for the functional ${\int }_{\Omega }F(x,u,|\nabla u{|}^2)dx$. The integrand $F$ is convex and satisfies some ellipticity and boundedness assumptions. We also develop a new monotonicity formula and an $\epsilon$-regularity theorem for such stationary solutions with no restriction on their images. We then use the idea of quantitative stratification to show that the k-th strata of the singular set of such solutions are k-rectifiable.