TY - JOUR

T1 - Rigorous continuum limit for the discrete network formation problem

AU - Haskovec, Jan

AU - Kreusser, Lisa Maria

AU - Markowich, Peter A.

N1 - KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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).

PY - 2019/5/17

Y1 - 2019/5/17

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/656469

UR - https://www.tandfonline.com/doi/full/10.1080/03605302.2019.1612909

UR - http://www.scopus.com/inward/record.url?scp=85066094646&partnerID=8YFLogxK

U2 - 10.1080/03605302.2019.1612909

DO - 10.1080/03605302.2019.1612909

M3 - Article

VL - 44

SP - 1159

EP - 1185

JO - Communications in Partial Differential Equations

JF - Communications in Partial Differential Equations

SN - 0360-5302

IS - 11

ER -