A nonautonomous chain rule in W1,p and BV

In this paper we consider the chain rule formula for compositions x bar right arrow F(u(x)) in the case when u has a Sobolev or BV regularity and F(x, z) is separately Sobolev, or BV, with respect to x and C (1) with respect to z. Our results extend to this nonautonomous case the results known for compositions x bar right arrow F(u(x)).