Sobolev spaces on arbitrary domains and semigroups generated by fractional Laplacian

2022/03/29 by Reinhard Farwig, Tsukasa Iwabuchi, Farwig, Reinhard +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in engineering #Primary 46E35 #Secondary 46F12

paper · pdf · doi:10.48550/arxiv.2203.15157

openalex publication_date 2022/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe a procedure to introduce Sobolev spaces and the semigroup generated by the fractional Dirichlet Laplacian on an arbitrary domain of \R^d. In particular, the well-definedness of the spaces of both non-homogeneous and homogeneous type together with their duality properties, embeddings, and Gagliardo-Nirenberg inequalities will be discussed. We also show the continuity and the smoothing property of the semigroup.

Related

• Fractional Laplacian, homogeneous Sobolev spaces and their realizations
• The semigroup generated by the Dirichlet Laplacian of fractional order
• Semigroups on generalized Sobolev spaces associated with Laplacians with applications Stochastic PDEs with singular boundary conditions
• Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces
• Embedding and extension results in Fractional Musielak-Sobolev spaces