{"ID":3053197,"CreatedAt":"2026-06-04T04:41:36.695875263Z","UpdatedAt":"2026-06-05T18:30:19.334466044Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.04129","arxiv_id":"2606.04129","title":"A duality approach to the dense graph limit for biological transportation networks","abstract":"We develop a duality-based formulation of the dense graph limit for a variational model of biological transportation networks, where edge conductivities balance pumping power against metabolic cost. In contrast to the pressure-based approach of our previous work, which required conductivities to be uniformly positive, the present formulation allows general nonnegative conductivity kernels. The kinetic energy is defined through a dual variational principle, which remains meaningful for degenerate integrable kernels and assigns infinite energy when the associated nonlocal Poisson problem is not solvable. Using this formulation, we prove $Γ$-convergence in the sense of Mosco of the semidiscrete network energies to a continuum energy on symmetric nonnegative kernels. The convergence is obtained in the natural $L^γ$ topology dictated by the metabolic term. The $Γ$-$\\liminf$ inequality follows directly from the dual formulation, while $Γ$-$\\limsup$ recovery sequences are constructed by positive regularization of the conductivity kernels.","short_abstract":"We develop a duality-based formulation of the dense graph limit for a variational model of biological transportation networks, where edge conductivities balance pumping power against metabolic cost. In contrast to the pressure-based approach of our previous work, which required conductivities to be uniformly positive,...","url_abs":"https://arxiv.org/abs/2606.04129","url_pdf":"https://arxiv.org/pdf/2606.04129v1","authors":"[\"Nuno J. Alves\",\"Jan Haskovec\"]","published":"2026-06-02T18:39:14Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}