Isometric Lattice Homomorphisms between Sobolev Spaces

TL;DR

This paper characterizes when isometric lattice homomorphisms between Sobolev spaces are induced by rigid motions, leading to domain congruence, thus linking geometric transformations with functional space isometries.

Contribution

It provides sufficient conditions under which Sobolev space isometries correspond to geometric rigid motions, extending previous results to more general settings.

Findings

Isometries correspond to rigid motions under certain conditions

Domains are shown to be congruent when Sobolev space isometries exist

Generalizations of the main results are established

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.