Existence and uniqueness of solutions to the norm minimum problem on digraphs
Chong Wang · Open Mathematics · 2022
Abstract In this article, based on the path homology theory of digraphs, which has been initiated and studied by Grigor’yan, Lin, Muranov, and Yau, we prove the existence and uniqueness of solutions to the problem ∥ w ∥ = min u ∈ Ω 2 ( G ) , u ≠ 0 1 2 ∥ ∂ u − w ∥ 2 2 + ∣ u ∣ 1 \parallel w\parallel =\mathop{\min }\limits_{u\in {\Omega }_{2}\left(G),u e 0}\left\{\phantom{\rule[-1.25em]{}{0ex}},\frac{1}{2}{\parallel \partial u-w\parallel }_{2}^{2}+{| u| }_{1}\right\} for w ∈ H 1 ( G ) w\in {H}_{1}\left(G) and any digraph G G generated by squares and triangles belonging to the same cluster.