A note for W^{1,p}(V) and W_0^{1,p}(V) on a locally finite graph

TL;DR

This paper explores Sobolev spaces on locally finite graphs, revealing that W^{1,p}(V) and W_0^{1,p}(V) are generally not equivalent, contrasting with Euclidean space, impacting variational methods for PDEs on graphs.

Contribution

It demonstrates the non-equivalence of Sobolev spaces W^{1,p}(V) and W_0^{1,p}(V) on locally finite graphs, a novel insight differing from classical Euclidean results.

Findings

W^{1,p}(V) is not equivalent to W_0^{1,p}(V) on locally finite graphs

This non-equivalence contrasts with Euclidean space properties

Implications for variational methods on graph-based PDEs

Abstract

In this paper, we investigate the Sobolev spaces W^{1,p}(V) and W_0^{1,p}(V) on a locally finite graph G=(V,E), which are fundamental tools when we apply the variational methods to partial differential equations on graphs. As a key contribution of this note, we show that in general, W^{1,p}(V) is not equivalent to W_0^{1,p}(V) on locally finite graphs, which is different from the situation on Euclidean space R^N.