论文标题
$σ$ - 局部紧凑的设置中的因果变异原理：最小化的存在
Causal Variational Principles in the $σ$-Locally Compact Setting: Existence of Minimizers
论文作者
论文摘要
我们证明了第二个可计数，紧凑的Hausdorff空间的因果变异原理的最小化。此外，得出了相应的欧拉 - 拉格朗日方程。该方法首先证明存在因果变异原理的最小化物，仅限于较低半连续拉格朗日的紧凑子集。通过紧凑的子集耗尽潜在的拓扑空间并重新缩放相应的最小化器，我们获得了一个序列，该序列模糊地收敛到可能无限总体积的常规Borel量度。结果表明，对于紧凑范围的连续拉格朗日人，该度量解决了Euler-Lagrange方程。此外，我们证明，在紧凑的支持的变化下，构造的度量是一种最小化。在其他假设下，证明该措施是有限体积变化的最小化器。我们最终将结果扩展到在熵中腐烂的连续拉格朗日人。
We prove the existence of minimizers of causal variational principles on second countable, locally compact Hausdorff spaces. Moreover, the corresponding Euler-Lagrange equations are derived. The method is to first prove the existence of minimizers of the causal variational principle restricted to compact subsets for a lower semi-continuous Lagrangian. Exhausting the underlying topological space by compact subsets and rescaling the corresponding minimizers, we obtain a sequence which converges vaguely to a regular Borel measure of possibly infinite total volume. It is shown that, for continuous Lagrangians of compact range, this measure solves the Euler-Lagrange equations. Furthermore, we prove that the constructed measure is a minimizer under variations of compact support. Under additional assumptions, it is proven that this measure is a minimizer under variations of finite volume. We finally extend our results to continuous Lagrangians decaying in entropy.
