On Sobolev spaces of bounded subanalytic manifolds

• Published in: Mathematische annalen 2024, Vol.390 (2), p.2413-2457
• Main Author: Valette, Guillaume
• Format: Article
• Language: English
• Subjects: Dirichlet problem; Embedding; Laplace equation; Manifolds (mathematics); Mathematics; Mathematics and Statistics; Operators (mathematics); Sobolev space; Theorems
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-024-02810-2

We focus on the Sobolev spaces of bounded subanalytic submanifolds of R n . We prove that if M is such a manifold then the space C 0 ∞ ( M ) is dense in W 1 , p ( M , ∂ M ) (the kernel of the trace operator) for all p ≤ p M , where p M is the codimension in M of the singular locus of M ¯ \ M (which is always at least 2). In the case where M is normal, i.e. when B ( x 0 , ε ) ∩ M is connected for every x 0 ∈ M ¯ and ε > 0 small, we show that C ∞ ( M ¯ ) is dense in W 1 , p ( M ) for all such p . This yields some duality results between W 1 , p ( Ω , ∂ Ω ) and W - 1 , p ′ ( Ω ) in the case where 1 < p ≤ p Ω and Ω is a bounded subanalytic open subset of R n . As a byproduct, we deduce uniqueness of the (weak) solution of the Dirichlet problem associated with the Laplace equation. We then prove a version of Sobolev’s Embedding Theorem for subanalytic bounded manifolds, show Gagliardo–Nirenberg’s inequality (for all p ∈ [ 1 , ∞ ) ), and derive some versions of Poincaré–Friedrichs’ inequality. We finish with a generalization of Morrey’s Embedding Theorem.