标题： 关于非紧致和等变情形下因果变分原理的极小化器的结构 显示英文标题

标题： On the Structure of Minimizers of Causal Variational Principles in the Non-Compact and Equivariant Settings

摘要： 我们推导了非紧致情形下带约束的因果变分原理极小化问题的欧拉-拉格朗日方程，可能还涉及对称性约束。首先考虑一阶变分，我们证明极小化测度支撑于一个超平面与某个二阶齐次函数的水平集的交集上。此外，我们进行二阶变分以得到表示作用量二次部分的紧算子是半正定的。证明的关键在于对凸集上的变分原理进行了拉格朗日乘数法的一个精细调整。 显示英文摘要

摘要： We derive the Euler-Lagrange equations for minimizers of causal variational principles in the non-compact setting with constraints, possibly prescribing symmetries. Considering first variations, we show that the minimizing measure is supported on the intersection of a hyperplane with a level set of a function which is homogeneous of degree two. Moreover, we perform second variations to obtain that the compact operator representing the quadratic part of the action is positive semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange multiplier method to variational principles on convex sets.