Tangential differential calculus and functional analysis on a \(C^{1,1}\) submanifold (Q2708675)

This article deals with a completely intrinsic approach to Sobolev spaces on bounded open domains \(\omega\) in \(C^{1,1}\)-submanifolds of codimension one of the Euclidean space \(\mathbb{R}^N\); more precisely the author considers bounded open domains in the boundary \(\Gamma\) of an open subset \(\Omega \subseteq \mathbb{R}^N\) of the class \(C^{1,1}\); his method is based on a ``marriage'' of the notion of tangential derivative and the oriented distance function \(b_\Omega(x)= d(x,\Omega)- d(x,\mathbb{R}^N \setminus\Omega):\) \(\mathbb{R}^n\to \mathbb{R}\) and leads to the definition of Sobolev spaces, which is equivalent to the usual one through local \(C^{1,1}\) diffeomorphisms.NEWLINENEWLINENEWLINEThe basic results are the theorems about isomorphism between the \(L^p\)-functions on \(\omega\) and on its Euclidean neighbourhood, density theorems for \(W^{1,p}(\omega)\), \(1\leq p< \infty\), theorems about the sense of the tangential gradient for \(C^{0,1} (\omega)\) functions, and others.NEWLINENEWLINENEWLINEIt is claimed that this approach is essentially simpler than usual one for the associated calculus and functional analysis; this is illustrated with the first completely intrinsic proof of the well known Korn's inequality.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00042].