Isometric Lattice Homomorphisms between Sobolev Spaces

Abstract

Given bounded domains ${\Omega }_1$ and ${\Omega }_2$ in $\mathds{R}^N$ and an isometry $T$ from $W^{1,p}({\Omega }_1)$ to $W^{1,p}({\Omega }_2)$, we give sufficient conditions ensuring that $T$ corresponds to a rigid motion of the space, i.e., $Tu=\pm (u\circ \xi )$ for an isometry $\xi$, and that the domains are congruent. More general versions of the involved results are obtained along the way.

A mistake in the case p=2 has been fixed. The article has been thoroughly rewritten. 21 pages

References in corpus (2)

• The Relative Capacity
• Lattice Homomorphisms between Sobolev Spaces