Abstract:We address the problem of determining the weakly continuous polynomials for sequences of functions that satisfy general linear first-order differential constraints. We prove that wedge products are weakly continuous when the differential constraints are given by exterior derivatives. This is sufficient for reproducing the Div-Curl Lemma of Murat and Tartar, the null Lagrangians in the calculus of variations and the weakly continuous polynomials for Maxwell’s equations. This result was derived independently by Tartar who stated it in a recent survey article . Our proof is explicit and uses the Hodge decomposition.
- John M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403. MR 475169, DOI 10.1007/BF00279992
- Bernard Dacorogna, Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Mathematics, vol. 922, Springer-Verlag, Berlin-New York, 1982. MR 658130
- R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82 (1983), no. 1, 27–70. MR 684413, DOI 10.1007/BF00251724
- Dominic G. B. Edelen, The null set of the Euler-Lagrange operator, Arch. Rational Mech. Anal. 11 (1962), 117–121. MR 150623, DOI 10.1007/BF00253934
- L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
- Luc Tartar, The compensated compactness method applied to systems of conservation laws, Systems of nonlinear partial differential equations (Oxford, 1982) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 111, Reidel, Dordrecht, 1983, pp. 263–285. MR 725524 —, Oscillations in nonlinear partial differential equations, Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, R.I., 1986.
- © Copyright 1987 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 303 (1987), 609-618
- MSC: Primary 58A14; Secondary 35A30, 58C07, 58G30
- DOI: https://doi.org/10.1090/S0002-9947-1987-0902788-8
- MathSciNet review: 902788