Rigorous continuum limit for the discrete network formation problem
Communications in Partial Differential Equations
Taylor & Francis
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Haskovec, J., Kreusser, L., & Markowich, P. (2019). Rigorous continuum limit for the discrete network formation problem. Communications in Partial Differential Equations, 44 (11), 1159-1185. https://doi.org/10.1080/03605302.2019.1612909
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.
LMK was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016516/1 and the German National Academic Foundation (Studienstiftung des Deutschen Volkes).
External DOI: https://doi.org/10.1080/03605302.2019.1612909
This record's URL: https://www.repository.cam.ac.uk/handle/1810/295978
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