arXiv subjects
Publications and source records attributed to Markus Biegert.
Given bounded domains $Ω_1$ and $Ω_2$ in $\mathds{R}^N$ and an isometry $T$ from $W^{1,p}(Ω_1)$ to $W^{1,p}(Ω_2)$, we give sufficient conditions ensuring that $T$ corresponds to a rigid motion of the space, i.e., $Tu = \pm (u \circ ξ)$ for an isometry $ξ$, and that the domains are congruent. More general versions of the involved results are obtained along the way.
The purpose of this article is to define a capacity on certain topological measure spaces $X$ with respect to certain function spaces $V$ consisting of measurable functions. In this general theory we will not fix the space $V$ but we emphasize that $V$ can be the classical Sobolev space $W^{1,p}(Ω)$, the classical Orlicz-Sobolev space $W^{1,Φ}(Ω)$, the Hajłasz-Sobolev space $M^{1,p}(Ω)$, the Musielak-Orlicz-Sobolev space (or generalized Orlicz-Sobolev space) and many other spaces. Of particular interest is the space $V:=\tW^{1,p}(Ω)$ given as the closure of $W^{1,p}(Ω)\cap C_c(\overlineΩ)$ in $W^{1,p}(Ω)$. In this case every function $u\in V$ (a priori defined only on $Ω$) has a trace on the boundary $\partialΩ$ which is unique up to a $\Cap_{p,Ω}$-polar set.
We show that every vector lattice homomorphism $T$ between Sobolev spaces can be represented by a composition and a multiplication, that is, $T$ is of the form $Tu(x)=u(h(x))g(x)$ for quasi every/almost every $x$ and all $u$.
The purpose of this article is to introduce the relative $p$-capacity $\Cap_{p,Ω}$ with respect to an open set $Ω$ in $\IR^N$. It is a Choquet capacity on the closure of $Ω$ and extends the classical $p$-capacity $\Cap_p$ in the sense that $\Cap_{p,Ω}=\Cap_p$ if $Ω=\IR^N$. The importance of the relative $p$-capacity stems from the fact that a large class of Sobolev functions defined on a 'bad domain' admit a trace on the boundary $\partialΩ$ which is then unique up to $\Cap_{p,Ω}$-polar set. As an application we prove a characterization of $W^{1,p}_0(Ω)$ for open sets $Ω\subset\IR^N$.
We prove under a weak smoothness condition that two Riemannian manifold are isomorphic if and only there exists an order isomorphism which intertwines with the Dirichlet type heat semigroups on the manifolds.