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, $W^{1,p}(V)\neq W_0^{1,p}(V)$ on locally finite graphs, which is different from the situation on Euclidean space $\mathbb{R}^N$.
Yulu TIAN
,
Liang ZHAO
. A NOTE FOR $W^{1,P}(V)$ AND $W_{0}^{1,P}(V)$ ON A LOCALLY FINITE GRAPH[J]. Acta mathematica scientia, Series B, 2025
, 45(5)
: 2135
-2141
.
DOI: 10.1007/s10473-025-0517-8
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