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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 Müller (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.

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Notes

  1. The standard definition requires an additional factor \((-1)^{p+q}\), where \(p\) and \(q\) denote positions of \((p)\) and \((q)\) in an appropriate ordering of the elements of \(I^m_s\) and \(I^n_s\), respectively. However, as this factor plays no role in the proof (and could be absorbed into the corresponding constant, anyway), we omit it here.

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Acknowledgments

The work was conducted during repeated mutual visits of the authors at the Universities of Cologne and Warsaw and at the Institute of Information Theory and Automation in Prague. The hospitality and support of all these institutions is gratefully acknowledged. The work of AK was supported by the Polish Ministry of Science grant no. N N201 397837 (years 2009-2012). SK and MK were supported by the AVCR-DAAD project CZ01-DE03/2013-2014 (DAAD project id. 56269992), and MK acknowledges support by the grants P201/10/0357 and P105/11/0411(GA ČR).

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Correspondence to Martin Kružík.

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Communicated by J. Ball.

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Kałamajska, A., Krömer, S. & Kružík, M. Sequential weak continuity of null Lagrangians at the boundary. Calc. Var. 49, 1263–1278 (2014). https://doi.org/10.1007/s00526-013-0621-9

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  • DOI: https://doi.org/10.1007/s00526-013-0621-9

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