Langer, Christoph (2023) Causal variational principles in the infinite-dimensional setting: Existence of minimizers. ADVANCES IN CALCULUS OF VARIATIONS, 16 (2). pp. 299-336. ISSN 1864-8258, 1864-8266
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We provide a method for constructing (possibly non-trivial) measures on non-locally compact Polish subspaces of infinite-dimensional separable Banach spaces which, under suitable assumptions, are minimizers of causal variational principles in the non-locally compact setting. Moreover, for non-trivial minimizers the corresponding Euler-Lagrange equations are derived. The method is to exhaust the underlying Banach space by finite-dimensional subspaces and to prove existence of minimizers of the causal variational principle restricted to these finite-dimensional subsets of the Polish space under suitable assumptions on the Lagrangian. This gives rise to a corresponding sequence of minimizers. Restricting the resulting sequence to countably many compact subsets of the Polish space, by considering the resulting diagonal sequence, we are able to construct a regular measure on the Borel algebra over the whole topological space. For continuous Lagrangians of bounded range, it can be shown that, under suitable assumptions, the obtained measure is a (possibly non-trivial) minimizer under variations of compact support. Under additional assumptions, we prove that the constructed measure is a minimizer under variations of finite volume and solves the corresponding Euler-Lagrange equations. Afterwards, we extend our results to continuous Lagrangians vanishing in entropy. Finally, assuming that the obtained measure is locally finite, topological properties of spacetime are worked out and a connection to dimension theory is established.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | Causal variational principles; existence theory; topological measure theory |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Petra Gürster |
| Date Deposited: | 15 May 2024 07:54 |
| Last Modified: | 15 May 2024 07:54 |
| URI: | https://pred.uni-regensburg.de/id/eprint/45767 |
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