# Manifold constrained non-uniformly elliptic problems

## TL;DR

This paper studies the regularity of minimizers for a class of constrained variational problems with non-uniformly elliptic energies, introducing new methods to analyze singular sets and partial regularity.

## Contribution

It develops intrinsic techniques for partial regularity and singular set estimates in non-uniformly elliptic problems with sphere constraints, expanding the understanding of such variational integrals.

## Key findings

- Established partial regularity results for minimizers.
- Provided estimates for the size of singular sets using Hausdorff measures.
- Developed comparison methods with capacities for non-uniformly elliptic problems.

## Abstract

We consider the problem of minimizing variational integrals defined on \cc{nonlinear} Sobolev spaces of competitors taking values into the sphere. The main novelty is that the underlying energy features a non-uniformly elliptic integrand exhibiting different polynomial growth conditions and no homogeneity. We develop a few intrinsic methods aimed at proving partial regularity of minima and providing techniques for treating larger classes of similar constrained non-uniformly elliptic variational problems. In order to give estimates for the singular sets we use a general family of Hausdorff type measures following the local geometry of the integrand. A suitable comparison is provided with respect to the naturally associated capacities.