A simple example of the weak discontinuity of $f\mapsto \int \det \nabla f$

TL;DR

This paper demonstrates the weak discontinuity of the functional involving the determinant of the gradient, even for well-behaved conformal diffeomorphisms, highlighting subtle issues in the calculus of variations.

Contribution

It provides a simple example of weak discontinuity of the determinant functional using conformal diffeomorphisms, emphasizing the complexity of lower-semicontinuity in Sobolev spaces.

Findings

Conformal diffeomorphisms can converge weakly to a constant in Sobolev spaces.

The determinant functional is not weakly lower-semicontinuous even for smooth, injective functions.

Weak discontinuity persists despite restrictions to well-behaved function spaces.

Abstract

Verifying lower-semicontinuity of integral functionals in the weak topology of Sobolev spaces is a central theme in the calculus of variations. For integral functionals with $p$-growth, quasiconvexity is a necessary condition for weak lower-semicontinuity in $W^{1,p}$, but is only sufficient if some additional conditions are met.The standard functional showing the necessity of additional conditions is $f\mapsto \int_\Omega \det \nabla f$, which fails to be weakly lower-semicontinuous. However, the common examples showing this failure are non-injective and have a lot of shear. The aim of this short note is to point out that a known sequence of conformal diffeomorphisms of the $d$-dimensional unit ball that converges weakly to a constant in $W^{1,d}$, exemplifies the weak discontinuity of this functional even when restricting a space to functions which are "as nice as possible".