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Causal Variational Principles in the $σ$-Locally Compact Setting: Existence of Minimizers
- Publication Year :
- 2020
- Publisher :
- arXiv, 2020.
-
Abstract
- 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.<br />Comment: 31 pages, LaTeX, small improvements (published version supplemented by appendix)
- Subjects :
- Pure mathematics
Applied Mathematics
010102 general mathematics
FOS: Physical sciences
Mathematical Physics (math-ph)
01 natural sciences
Functional Analysis (math.FA)
010101 applied mathematics
Mathematics - Functional Analysis
Mathematics - Classical Analysis and ODEs
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
Locally compact space
0101 mathematics
Analysis
Geometry and topology
Mathematical Physics
Mathematics
Subjects
Details
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....e7258eb0a7da6515dfb07eba4ac55dcd
- Full Text :
- https://doi.org/10.48550/arxiv.2002.04412