Causal variational principles in the σ-locally compact setting: Existence of minimizers
-
Felix Finster
and
Christoph Langer
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.
Funding statement: Christoph Langer gratefully acknowledges generous support by the “Studienstiftung des deutschen Volkes.”
Acknowledgements
We would like to thank Magdalena Lottner, Marco Oppio, Johannes Wurm and the unknown referee for helpful comments on the manuscript.
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Articles in the same Issue
- Frontmatter
- Michell truss type theories as a Γ-limit of optimal design in linear elasticity
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- Uniqueness and nonuniqueness of limits of Teichmüller harmonic map flow
- Regularity for minimizers of a class of non-autonomous functionals with sub-quadratic growth
- On a comparison principle for Trudinger’s equation
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- Causal variational principles in the σ-locally compact setting: Existence of minimizers
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