In the non-convex calculus of variations, in the context of non-linear elasticity, the following classes of mappings $u:\Omega\to\mathbb{R}^n$, $\Omega\subset\mathbb{R}^n$, were introduced by John Ball [B]: $$ A_{p,q}=\{u:\, \nabla u\in L^p,\ \ \operatorname{adj}\nabla u\in L^q\}, \quad p>\frac{n}{n-1},\ q\geq\frac{p}{p-1}. $$ $$ A_{p,q}^+=\{u\in A_{p,q}:\, \operatorname{det}\nabla u>0\ a.e.\}. $$ This class of mappings was investigated by Šverák [S] and you can find more information by searching papers on MathSciNet that cite papers by Ball and Sverak.
Question. What is known about density of smooth functions in $A_{p,q}$ and $A_{p,q}^+$ spaces?
I am interested both in density in the weak and strong topology, but mostly in the strong topology.
[B] J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1976/77), no. 4, 337–403.
[S] V. Šverák, Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988), 105–127.
