An exterior differential calculus, based on Lagrangian-mollified weak derivatives and osculating vacua, is constructed for non-smooth causal variational principles, with cohomology, Stokes and Gauss theorems, and worked lattice and discrete examples.
Complexes from complexes
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abstract
This paper is concerned with the derivation and properties of differential complexes arising from a variety of problems in differential equations, with applications in continuum mechanics, relativity, and other fields. We present a systematic procedure which, starting from well-understood differential complexes such as the de Rham complex, derives new complexes and deduces the properties of the new complexes from the old. We relate the cohomology of the output complex to that of the input complexes and show that the new complex has closed ranges, and, consequently, satisfies a Hodge decomposition, Poincar\'e type inequalities, well-posed Hodge-Laplacian boundary value problems, regular decomposition, and compactness properties on general Lipschitz domains.
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2026 1verdicts
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The $\mathcal{L}$-Calculus for Causal Variational Principles: An Exterior Differential Calculus on Non-Smooth Spaces
An exterior differential calculus, based on Lagrangian-mollified weak derivatives and osculating vacua, is constructed for non-smooth causal variational principles, with cohomology, Stokes and Gauss theorems, and worked lattice and discrete examples.