Elliptic methods for solving the linearized field equations of causal variational principles

The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted L...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations Vol. 61; no. 4
Main Authors: Finster, Felix, Lottner, Magdalena
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2022
Springer Nature B.V
Subjects:
Analysis
Calculus
Calculus of Variations and Optimal Control; Optimization
Control
Differential calculus
Elliptic functions
Fermions
Function space
Hilbert space
Linearization
Mathematical analysis
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operators (mathematics)
Partial differential equations
Systems Theory
Theoretical
Variational principles
46E35
49S05
35J50
49Q20
58C35
ISSN:0944-2669, 1432-0835
Online Access:Get full text
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Summary:The existence theory is developed for solutions of the inhomogeneous linearized field equations for causal variational principles. These equations are formulated weakly with an integral operator which is shown to be bounded and symmetric on a Hilbert space endowed with a suitably adapted weighted L 2 -scalar product. Guided by the procedure in the theory of linear elliptic partial differential equations, we use the spectral calculus to define Sobolev-type Hilbert spaces and invert the linearized field operator as an operator between such function spaces. The uniqueness of the resulting weak solutions is analyzed. Our constructions are illustrated in simple explicit examples. The connection to the causal action principle for static causal fermion systems is explained.
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ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-022-02237-0