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

TL;DR

This paper establishes partial regularity and stratification results for stationary solutions of a convex variational functional mapping into Riemannian manifolds, introducing new monotonicity and regularity theorems.

Contribution

It proves partial regularity and rectifiability of the singular set for stationary solutions of convex functionals with no image restrictions, using new monotonicity and stratification techniques.

Findings

Partial regularity of stationary solutions is established.

The singular set is shown to be k-rectifiable.

A new monotonicity formula and epsilon-regularity theorem are developed.

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.