Approximation in Sobolev spaces of nonlinear expressions involving the gradient

We investigate a problem of approximation of a large class of nonlinear expressionsf(x, u, ∇u), including polyconvex functions. Hereu: Ω→R ^m , Ω⊂R ⁿ , is a mapping from the Sobolev spaceW ^1,p . In particular, whenp=n, we obtain the approximation by mappings which are continuous, differentiable a.e. and, if in additionn=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 off(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. The first author was supported by KBN grant no. 2-PO3A-055-14, and by a scholarship from the Swedish Institute. The second author was supported by Research Project CEZ J13/98113200007 and grants GAČR 201/97/1161 and GAUK 170/99. This research originated during the stay of both authors at the Max-Planck Institute for Mathematics in the Sciences in Leipzig, 1998, and completed during their stay at the Mittag-Leffler Institute, Djursholm, 1999. They thank the institutes for the support and the hospitality.