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Keywords:
Sobolev spaces; minors of the Jacobi matrix; weak and strong convergence; cartesian currents
Sobolev spaces; minors of the Jacobi matrix; weak and strong convergence; cartesian currents
Summary:
The paper investigates the nonlinear function spaces introduced by Giaquinta, Modica and Souček. It is shown that a function from $\operatorname{Cart}^p(\Omega ,\bold R^m)$ is approximated by $\Cal C ^1$ functions strongly in $\Cal A^q(\Omega ,\bold R^m)$ whenever $q<p$. An example is shown of a function which is in $\operatorname{cart}^p(\Omega ,\bold R^2)$ but not in $\operatorname{cart}^p(\Omega ,\bold R^2)$.
References:
[1] Giaquinta M., Modica G., Souček J.: Cartesian currents, weak dipheomorphisms and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 106 (1989), 97-159. {Erratum and addendum}. Arch. Rat. Mech. Anal. 109 (1990), 385-592. MR 0980756
[2] Giaquinta M., Modica G., Souček J.: Cartesian currents and variational problems for mappings into spheres. Annali S.N.S. Pisa 16 (1989), 393-485. MR 1050333
[3] Giaquinta M., Modica G., Souček J.: The Dirichlet energy of mappings with values into the sphere. Manuscripta Math. 65 (1989), 489-507. MR 1019705
[4] Giaquinta M., Modica G., Souček J.: The Dirichlet integral for mappings between manifolds: Cartesian currents and homology. Università di Firenze, preprint, 1991. MR 1183409
[5] V. Šverák: Regularity properties of deformations with finite energy. Arch. Rat. Mech. Anal. 100 (1988), 105-127. MR 0913960
[6] W.P. Ziemer: Weakly Differentiable Functions. Sobolev Spaces and Function of Bounded Variation. Graduate Text in Mathematics 120, Springer-Verlag, 1989. MR 1014685
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