Some singular minimizers in low dimensions in the calculus of variations

TL;DR

This paper constructs a specific example of a singular minimizer for a smooth, convex variational problem in three dimensions, demonstrating the existence of such minimizers in low-dimensional settings.

Contribution

It provides the first explicit example of a singular minimizer in low dimensions for a smooth, convex functional in the calculus of variations.

Findings

Existence of a singular minimizer in three dimensions.

The minimizer is smooth away from a singular set.

The functional is uniformly convex and smooth.

Abstract

We construct a singular minimizing map ${\bf u}$ from $\mathbb{R}^3$ to $\mathbb{R}^2$ of a smooth uniformly convex functional of the form $\int_{B_1} F(D{\bf u})\,dx$.