## 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.

## Notes

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.

## References

Alibert, J.J., Bouchitté, G.: Non-uniform integrability and generalized Young measures. J. Convex Anal.

**4**(1), 129–147 (1997)Ball, J.M.: A version of the fundamental theorem for young measures. In M. Rascle, D. Serre, and M. Slemrod, editors, PDEs and continuum models of phase transitions. Proceedings of an NSF-CNRS joint seminar held in Nice, France, January 18–22, 1988, volume 344 of Lect. Notes Phys., pages 207–215. Springer, Berlin (1989)

Ball, J.M., Murat, F.: \(W^{1, p}\)-quasiconvexity and variational problems for multiple integrals. J. Funct. Anal.

**58**(3), 225–253 (1984)Ball, J.M., Marsden, J.E.: Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Ration. Mech. Anal.

**86**, 251–277 (1984)Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal.

**63**, 337–403 (1977)Ciarlet, P.G. , Gogu, R., Mardare, C.: A notion of polyconvex function on a surface suggested by nonlinear shell theory. C. R. Math. Acad. Sci. Paris 349(21–22):1207–1211 (2011)

Dacorogna, B.: Direct methods in the calculus of variations. 2nd ed. Applied Mathematical Sciences 78. Springer, Berlin (2008)

DiPerna, R.J., Majda, A.J.: Oscillations and concentrations in weak solutions of the incompressible fluid equations. Comm. Math. Phys.

**108**(4), 667–689 (1987)Dunford, N., Schwartz, J.T.: Linear operators. Part I: General theory. With the assistance of William G. Bade and Robert G. Bartle. Repr. of the orig., publ. 1959 by John Wiley & Sons Ltd., Paperback ed. Wiley Classics Library. New York etc., Wiley (1988)

Engelking, R.: General topology. Rev. and compl. ed. Sigma Series in Pure Mathematics, 6. Berlin: Heldermann Verlag. viii, 529 p. DM 148.00 (1989)

Fonseca, I.: Lower semicontinuity of surface energies. Proc. R. Soc. Edinb., Sect. A, 120(1–2):99–115 (1992)

Fonseca, I., Kružík, M.: Oscillations and concentrations generated by \({\cal A}\)-free mappings and weak lower semicontinuity of integral functionals. ESAIM, Control Optim. Calc. Var. 16(2):472–502 (2010)

Fonseca, I., Müller, S., Pedregal, P.: Analysis of concentration and oscillation effects generated by gradients. SIAM J. Math. Anal.

**29**(3), 736–756 (1998)Grabovsky, Y., Mengesha, T.: Direct approach to the problem of strong local minima in calculus of variations. Calc. Var. Partial Differ. Equ.

**29**(1), 59–83 (2007)Grabovsky, Y., Mengesha, T.: Erratum: “Direct approach to the problem of strong local minima in calculus of variations” [Calc. Var. Partial Differential Equations 29(1), 2007, pp. 59–83; mr2305477]. Calc. Var. Partial Differ. Equ.

**32**(3), 407–409 (2008)Kałamajska, A.: On lower semicontinuity of multiple integrals. Colloq. Math.

**74**(1), 71–78 (1997)Kałamajska, A., Kružík, M.: Oscillations and concentrations in sequences of gradients. ESAIM, Control Optim. Calc. Var. 14(1):71–104 (2008)

Kinderlehrer, D., Pedregal, P.: Characterizations of Young measures generated by gradients. Arch. Ration. Mech. Anal.

**115**(4), 329–365 (1991)Kinderlehrer, D., Pedregal, P.: Gradient Young measures generated by sequences in Sobolev spaces. J. Geom. Anal.

**4**(1), 59–90 (1994)Kristensen, J., Rindler, F.: Characterization of generalized gradient Young measures generated by sequences in \(W^{1,1}\) and BV. Arch. Ration. Mech. Anal.

**197**(2), 539–598 (2010)Kružík, M.: Quasiconvexity at the boundary and concentration effects generated by gradients. ESAIM, Control Optim. Calc. Var. (2013)

Kružík, M., Luskin, M.: The computation of martensitic microstructure with piecewise laminates. J. Sci. Comput.

**19**, 293–308 (2003)Kružík, M., Roubíček, T.: On the measures of DiPerna and Majda. Math. Bohem.

**122**(4), 383–399 (1997)Kružík, M., Roubíček, T.: Optimization problems with concentration and oscillation effects: relaxation theory and numerical approximation. Numer. Funct. Anal. Optim.

**20**(5–6), 511–530 (1999)Kufner, A., John, O., Fučík, S.: Function spaces. Monographs and Textsbooks on Mechanics of Solids and Fluids. Mechanics: Analysis. Leyden: Noordhoff International Publishing. (1977)

Mielke, A., Sprenger, P.: Quasiconvexity at the boundary and a simple variational formulation of Agmon’s condition. J. Elasticity

**51**(1), 23–41 (1998)Morrey, C.B.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pac. J. Math.

**2**, 25–53 (1952)Müller, S.: A surprising higher integrability property of mappings with positive determinant. Bull. Am. Math. Soc. New Ser.

**21**(2), 245–248 (1989)Müller, S.: Higher integrability of determinants and weak convergence in \(L^1\). J. Reine Angew. Math.

**412**, 20–34 (1990)Pedregal, P.: Parametrized measures and variational principles. Progress in Nonlinear Differential Equations and their Applications. 30. Basel: Birkhäuser. (1997)

Roubíček, T.: Relaxation in optimization theory and variational calculus. de Gruyter Series in Nonlinear Analysis and Applications. 4. Walter de Gruyter, Berlin (1997)

Šilhavý, M.: Equilibrium of phases with interfacial energy: a variational approach. J. Elasticity

**105**(1–2), 271–303 (2011)Šilhavý, M.: The Mechanics and Thermodynamics of Continuous Media. Texts and Monographs in Physics. Springer, Berlin (1997)

Sprenger, P.: Quasikonvexität am Rande und Null-Lagrange-Funktionen in der nichtkonvexen Variationsrechnung. PhD thesis, Universität Hannover (1996)

Tartar, L.: Mathematical tools for studying oscillations and concentrations: From Young measures to \(H\)-measures and their variants. Antonić, Nenad (ed.) et al., Multiscale problems in science and technology. Challenges to mathematical analysis and perspectives. Proceedings of the conference on multiscale problems in science and technology, Dubrovnik, Croatia, September 3–9, 2000. Springer, Berlin (2002)

Young, L.C.: Generalized curves and the existence of an attained absolute minimum in the calculus of variations. C. R. Soc. Sci. Varsovie

**30**, 212–234 (1937)

## 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).

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

Communicated by J. Ball.

## Rights and permissions

## About this article

### Cite this article

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

Received:

Accepted:

Published:

Issue Date:

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