Sequential weak continuity of null Lagrangians at the boundary

TL;DR

This paper investigates the weak* measure continuity of null Lagrangians at the boundary, characterizes null Lagrangians at the boundary in arbitrary dimensions, and discusses implications for weak continuity and lower semicontinuity in variational calculus.

Contribution

It provides a precise characterization of null Lagrangians at the boundary and establishes new weak lower semicontinuity results for integrands involving these Lagrangians.

Findings

Weak* measure continuity of null Lagrangians at the boundary.

Characterization of null Lagrangians at the boundary in arbitrary dimensions.

Counterexample showing limitations of higher integrability results.

Abstract

We show weak* in measures on $\bar\O$/ weak-$L^1$ sequential continuity of $u\mapsto f(x,\nabla u):W^{1,p}(\O;\R^m)\to L^1(\O)$, where $f(x,\cdot)$ is a null Lagrangian for $x\in\O$, it is a null Lagrangian at the boundary for $x\in\partial\O$ and $|f(x,A)|\le C(1+|A|^p)$. We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why $u\mapsto \det\nabla u:W^{1,n}(\O;\R^n)\to L^1(\O)$ fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians at the boundary. The paper closes with an example indicating that a well-known result on higher integrability of determinant \cite{Mue89a} need not necessarily extend to our setting. The notion of quasiconvexity at the boundary due to J.M. Ball and J. Marsden is central to our analysis.