A simple example of the weak discontinuity of $f\mapsto \int \det \nabla f$
Cy Maor
TL;DR
The paper investigates weak lower-semicontinuity of integral functionals in $W^{1,p}$, focusing on the determinant functional $I_{\operatorname{det}}(f)=\int_\Omega \det \nabla f$ in $W^{1,d}$. It demonstrates that even when restricting to embeddings, a simple conformal-diffeomorphism sequence on the unit ball, which converges weakly to zero in $W^{1,d}$, can drive $I_{\operatorname{det}}$ to fail weak lower-semicontinuity due to boundary concentration of the gradient energy. The main constructive tool is a family of Möbius transformations that map the unit ball onto a fixed ball while preserving orientation in a limiting sense, yielding $\int_{B_1(0)} \det \nabla f_n = -\operatorname{Vol}(B_1(0))$. The results underscore the sharpness of concentration phenomena and show that even within the class of embeddings, additional hypotheses beyond quasiconvexity are necessary for weak lower-semicontinuity; in particular, restricting to nonnegative Jacobians can recover semicontinuity.
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 $f\mapsto \int_Ω\det \nabla f$, 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".