Communications in Mathematical Physics Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems
Communications in Mathematical Physics Discrete Pluriharmonic Functions as Solutions of Linear Pluri-Lagrangian Systems
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数学物理中的通信离散多调和函数作为线性多拉格朗日系统的解
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通讯作者：
Y. Suris
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作者：
A. Bobenko;Y. Suris
复数拉格朗日系统是具有多维一致性的变分系统。这个概念起源于多谐函数理论，统计力学的z不变模型，追溯到诺特的变分对称性理论以及离散可积系统理论。一个d维复数拉格朗日问题可以描述如下：给定一个d维空间上的d型L， m > d，其系数依赖于一个由m个自变量组成的函数u（称为场），找出那些场u，这些场u对m维空间中的任何d维流形的作用泛函S =∫L提供临界点。研究离散二维线性多元拉格朗日系统，即具有二次拉格朗日量L的系统。这种作用是狄利克雷能量的离散模拟，其解称为离散多谐函数。我们用拉格朗日算子根据对角线对线性复拉格朗日系统进行分类。它们是通过星三角图的概括来描述的。还考虑了更一般的二次拉格朗日的例子。
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 mdimensional 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 S = ∫ L for any d-dimensional manifold 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.