Approximation in Sobolev spaces of nonlinear expressions involving the gradient

Abstract

We investigate a problem of approximation of a large class of nonlinear expressions f(x, u, ∇u), including polyconvex functions. Here u: Ω→R^m, Ω⊂Rⁿ, is a mapping from the Sobolev space W^1,p. In particular, when p=n, we obtain the approximation by mappings which are continuous, differentiable a.e. and, if in addition n=m, satisfy the Luzin condition. From the point of view of applications such mappings are almost as good as Lipschitz mappings. As far as we know, for the nonlinear problems that we consider, no natural approximation results were known so far. The results about the approximation of f(x, u, ∇u) are consequences of the main result of the paper, Theorem 1.3, on a very strong approximation of Sobolev functions by locally weakly monotone functions.