Weak and strong L^p-limits of vector fields with finitely many integer singularities in dimension n

2022/10/10 by Riccardo N. Caniato, Caniato, Riccardo, Filippo Gaia +1
Mathematics · #46N20 (primary) #58E15 (secondary) #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2210.04730

openalex publication_date 2022/10/10 · openalex created_date 2022/10/14 · openalex updated_date 2026/07/28

Abstract

For every given p∈ [1,+∞) and n∈ℕ with n≥ 1, the authors identify the strong L^p-closure L_ℤ^p(D) of the class of vector fields having finitely many integer topological singularities on a domain D which is either bi-Lipschitz equivalent to the open unit n-dimensional cube or to the boundary of the unit (n+1)-dimensional cube. Moreover, for every n∈ℕ with n≥ 2 the authors prove that L_ℤ^p(D) is weakly sequentially closed for every p∈ (1,+∞) whenever D is an open domain in ℝⁿ which is bi-Lipschitz equivalent to the open unit cube. As a byproduct of the previous analysis, a useful characterisation of such class of objects is obtained in terms of existence of a (minimal) connection for their singular set.

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