A uniqueness criterion and a counterexample to regularity in an incompressible variational problem

TL;DR

This paper establishes a uniqueness criterion for minimizers in an incompressible variational problem and constructs a counterexample showing regularity can fail even with convex functionals.

Contribution

It introduces a criterion ensuring uniqueness of minimizers under small pressure gradient conditions and provides a counterexample with a Lipschitz but non-smooth minimizer.

Findings

Unique global minimizer under small pressure gradient

Counterexample with Lipschitz but non-$C^1$ minimizer

Functional depends smoothly on $ abla u$ but discontinuously on $x$

Abstract

In this paper we consider the problem of minimizing functionals of the form $E(u)=\int_B f(x,\nabla u) \,dx$ in a suitably prepared class of incompressible, planar maps $u: B \rightarrow \mathbb{R}^2$. Here, $B$ is the unit disk and $f(x,\xi)$ is quadratic and convex in $\xi$. It is shown that if $u$ is a stationary point of $E$ in a sense that is made clear in the paper, then $u$ is a unique global minimizer of $E(u)$ provided the gradient of the corresponding pressure satisfies a suitable smallness condition. We apply this result to construct a non-autonomous, uniformly convex functional $f(x,\xi)$, depending smoothly on $\xi$ but discontinuously on $x$, whose unique global minimizer is the so-called $N-$covering map, which is Lipschitz but not $C^1$.