TY - JOUR

T1 - Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems

AU - Bobenko, A. I.

AU - Suris, Yu B.

N1 - Publisher Copyright:
© 2014, Springer-Verlag Berlin Heidelberg.

PY - 2015/5

Y1 - 2015/5

N2 - Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of variational symmetries going back to Noether and in the theory of discrete integrable systems. A d-dimensional pluri-Lagrangian problem can be described as follows: given a d-form L on an m-dimensional space, m > d, whose coefficients depend on a function u of m independent variables (called field), find those fields u which deliver critical points to the action functionals (RFormula presented.) for any d-dimensional manifold Σ in the m-dimensional space. We investigate discrete 2-dimensional linear pluri-Lagrangian systems, i.e., those with quadratic Lagrangians L. The action is a discrete analogue of the Dirichlet energy, and solutions are called discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems with Lagrangians depending on diagonals. They are described by generalizations of the star-triangle map. Examples of more general quadratic Lagrangians are also considered.

AB - Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of variational symmetries going back to Noether and in the theory of discrete integrable systems. A d-dimensional pluri-Lagrangian problem can be described as follows: given a d-form L on an m-dimensional space, m > d, whose coefficients depend on a function u of m independent variables (called field), find those fields u which deliver critical points to the action functionals (RFormula presented.) for any d-dimensional manifold Σ in the m-dimensional space. We investigate discrete 2-dimensional linear pluri-Lagrangian systems, i.e., those with quadratic Lagrangians L. The action is a discrete analogue of the Dirichlet energy, and solutions are called discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems with Lagrangians depending on diagonals. They are described by generalizations of the star-triangle map. Examples of more general quadratic Lagrangians are also considered.

UR - http://www.scopus.com/inward/record.url?scp=84925487174&partnerID=8YFLogxK

U2 - 10.1007/s00220-014-2240-5

DO - 10.1007/s00220-014-2240-5

M3 - Article

AN - SCOPUS:84925487174

SN - 0010-3616

VL - 336

SP - 199

EP - 215

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 1

ER -