Approximation in Sobolev spaces of nonlinear expressions involving the gradient
The aim of the paper under review is to approximate mappings \(u:\Omega\rightarrow {\mathbb{R}}^m\) from a Sobolev space \(W^{1,p}(\Omega)^m\) (\(1\leq p<\infty\)) on an open set \(\Omega\subset {\mathbb{R}}^n\) by a sequence \(\{ u_k\}_{k=1}^\infty\) of mappings with some good regularity conditions. In the main theorem it is shown that the approximating functions \(u_k\) can be taken such that \(\nabla u_k=0\) a.e. in the set where \(u_k\neq u\), and \(u_k\) locally weakly monotone in the sense of \textit{J. J. Mamfredi} [J. Geom. Anal. 4, 393-402 (1994; Zbl 0805.35013)]. This allows the authors to obtain the convergence \(\int_\Omega f(x,u_k,\nabla u_k) dx \to\int_\Omega f(x,u,\nabla u) dx\) for a large class of integrands, which include many interesting special cases, such as \(\det \nabla u_k \to \det \nabla u\) in \(L^q(\Omega)\). When \(p=n\), every mapping \(u_k\) is continuous, differentiable a.e. and satisfies the Lusin property (the image of any set of Lebesgue measure zero is also of Lebesgue measure zero), and it is proved that this result is sharp.
scientific article; zbMATH DE number 1987657
scientific article