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<p>In this paper, we investigate the Sobolev spaces <i>W</i><sup>1,<i>p</i></sup>(<i>V</i>) and <i>W</i><Stack> <sub>0</sub> <sup>1,<i>p</i></sup> </Stack>(<i>V</i>) on a locally finite graph <i>G</i> = (<i>V, E</i>), 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, <i>W</i><sup>1,<i>p</i></sup>(<i>V</i>) ≠ <i>W</i><Stack> <sub>0</sub> <sup>1,<i>p</i></sup> </Stack>(<i>V</i>) on locally finite graphs, which is different from the situation on Euclidean space ℝ<sup><i>N</i></sup>.</p>

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A note for W1,p(V) and W 0 1,p (V) on a locally finite graph

  • 期刊论文

  • 发表日期：

    2025年8月4日

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  • Yulu Tian,
  • Liang Zhao

摘要

In this paper, we investigate the Sobolev spaces W1,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, W1,p(V) ≠ W 0 1,p (V) on locally finite graphs, which is different from the situation on Euclidean space ℝN.