Abstract Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their Γ-convergence towards a continuum energy functional.

中文翻译：

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离散网络形成问题的严格连续统限制

摘要 受近期描述生物运输网络形成的论文的启发，我们研究了 Hu 和 Cai 提出的离散模型，该模型由受图上线性系统约束的能量消耗函数组成。对于空间二维矩形设置，我们证明了约束能量泛函的严格连续限制，因为底层图的节点数趋于无穷大，边长均匀地收缩为零。该证明基于将离散能量泛函重新表示为一系列积分泛函，并证明它们的 Γ-收敛于连续能量泛函。