Sequential weak continuity of null Lagrangians at the boundary

Abstract

We show weak* in measures on \(\bar{\Omega }\)/ weak-\(L^1\) sequential continuity of \(u\mapsto f(x,\nabla u):W^{1,p}(\Omega ;\mathbb{R }^m)\rightarrow L^1(\Omega )\), where \(f(x,\cdot )\) is a null Lagrangian for \(x\in \Omega \), it is a null Lagrangian at the boundary for \(x\in \partial \Omega \) 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 \(u\mapsto \det \nabla u:W^{1,n}(\Omega ;\mathbb{R }^n)\rightarrow L^1(\Omega )\) 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 by Muller (Bull. Am. Math. Soc. New Ser. 21(2): 245–248, 1989) 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.