Communications in Mathematical Sciences

Volume 5 (2007)

Number 1

Diffuse Interface Energies Capturing the Euler Number: Relaxation and Renomalization

Pages: 233 – 242

(Fast Communication)



Qiang Du

Chun Liu

Rolf Ryham

Xiaoqiang Wang


We introduce a set of new interfacial energies for approximating the Euler number of level surfaces in the phase field (diffuse-interface) representation. These new formulae have simpler forms than those studied earlier in Q. Du, C. Liu and X. Wang, Retrieving topological information for phase field models, and do not contain higher order derivatives of the phase field function. Theoretical justifications are provided via formal asymptotic analysis, and practical validations are performed through numerical experiments. Relaxation and renormalization schemes are also developed to improve the robustness of the new energy functionals.


Euler number; topological index; implicitly defined surfaces; phase field; diffuse interface; interfacial energy; surface tension; bending energy; vesicle membrane

2010 Mathematics Subject Classification

57M50, 74A50, 74S99, 92C05

Full Text (PDF format)