具有凸变分原理的离散可积方程
On discrete integrable equations with convex variational principles
摘要：
本文研究了由离散可积四元方程驱动的离散拉普拉斯型方程的变分结构。特别是，我们解释了为什么现实条件，我们认为应该是所有的合理的，我们得到充分的条件（往往是必要的）标签的边缘下，相应的广义离散作用功能是凸的。凸性是讨论Dirichlet边值问题解的存在唯一性的重要工具。此外，我们研究的组合数据允许凸作用泛函的离散拉普拉斯型方程，实际上是由离散可积四元方程，我们提出了如何对应于（Q3）的方程和泛函有关的圆图案。
英文摘要：
We investigate the variational structure of discrete Laplace-type equations that are motivated by discrete integrable quad-equations. In particular, we explain why the reality conditions we consider should be all that are reasonable, and we derive sufficient conditions (that are often necessary) on the labeling of the edges under which the corresponding generalized discrete action functional is convex. Convexity is an essential tool to discuss existence and uniqueness of solutions to Dirichlet boundary value problems. Furthermore, we study which combinatorial data allow convex action functionals of discrete Laplace-type equations that are actually induced by discrete integrable quad-equations, and we present how the equations and functionals corresponding to (Q3) are related to circle patterns.
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