非紧等变环境下因果变分原理极小元的结构
On the Structure of Minimizers of Causal Variational Principles in the Non-Compact and Equivariant Settings
摘要：
在非紧的约束条件下，我们导出了因果变分原理的极小值的欧拉-拉格朗日方程，可能规定了对称性。考虑到第一变分，我们证明了极小化测度是支持在一个超平面与一个函数的水平集，这是齐次二的交集。此外，我们进行第二次变分，以获得表示二次部分的作用的紧算子是半正定的。证明的关键成分是拉格朗日乘子法对凸集上的变分原理的微妙适应。
英文摘要：
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.
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