A duality approach to the dense graph limit for biological transportation networks

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 $\Gamma$-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 $L^\gamma$ topology dictated by the metabolic term. The $\Gamma$-$\liminf$ inequality follows directly from the dual formulation, while $\Gamma$-$\limsup$ recovery sequences are constructed by positive regularization of the conductivity kernels.