Hodge Theory for Linearized Boundary-Value Problems on General Geometric Structures
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914622320148480 |
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| author | Leder, Roee |
| author_facet | Leder, Roee |
| contents | We develop a framework that systematically casts solvability and uniqueness conditions for linearized overdetermined boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the underlying geometric structure and provides tools to study the resulting cohomology explicitly. To achieve this generality, we develop the notion of an elliptic pre-complex, which generalizes the machineries of Hodge theory to sequences of Douglis--Nirenberg systems in the pseudodifferential calculus of boundary-value problems that interact through Green's formulae, the notion of overdetermined ellipticity, and a condition we call the order-reduction property, which relaxes the rigid requirement that the sequence form a cochain complex. This property typically arises from linearized symmetries and constraints, as we demonstrate through several geometric examples that have long resisted analysis, including exterior covariant derivatives, the Killing and Hessian equations, and the Riemann curvature equations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_18494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge Theory for Linearized Boundary-Value Problems on General Geometric Structures Leder, Roee Differential Geometry Mathematical Physics Analysis of PDEs Primary 58J10, 35N25, 53C21, Secondary 58J32, 35S15, 58A14 We develop a framework that systematically casts solvability and uniqueness conditions for linearized overdetermined boundary-value problems into cohomological terms. The theory is designed to be applicable without assumptions on the underlying geometric structure and provides tools to study the resulting cohomology explicitly. To achieve this generality, we develop the notion of an elliptic pre-complex, which generalizes the machineries of Hodge theory to sequences of Douglis--Nirenberg systems in the pseudodifferential calculus of boundary-value problems that interact through Green's formulae, the notion of overdetermined ellipticity, and a condition we call the order-reduction property, which relaxes the rigid requirement that the sequence form a cochain complex. This property typically arises from linearized symmetries and constraints, as we demonstrate through several geometric examples that have long resisted analysis, including exterior covariant derivatives, the Killing and Hessian equations, and the Riemann curvature equations. |
| title | Hodge Theory for Linearized Boundary-Value Problems on General Geometric Structures |
| topic | Differential Geometry Mathematical Physics Analysis of PDEs Primary 58J10, 35N25, 53C21, Secondary 58J32, 35S15, 58A14 |
| url | https://arxiv.org/abs/2504.18494 |