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Nonlocal trace spaces and extension results for nonlocal calculus

2022; Elsevier BV; Volume: 282; Issue: 12 Linguagem: Inglês

10.1016/j.jfa.2022.109453

1096-0783

Qiang Du, Xiaochuan Tian, Cory Wright, Yue Yu,

Nonlinear Partial Differential Equations

For a given Lipschitz domain Ω, it is a classical result that the trace space of W1,p(Ω) is W1−1/p,p(∂Ω), namely any W1,p(Ω) function has a well-defined W1−1/p,p(∂Ω) trace on its codimension-1 boundary ∂Ω and any W1−1/p,p(∂Ω) function on ∂Ω can be extended to a W1,p(Ω) function. In this work, we study function spaces for nonlocal Dirichlet problems involving integrodifferential operators with a finite range of nonlocal interactions, and provide a characterization of their trace spaces. For these nonlocal Dirichlet problems, the boundary conditions are normally imposed on a region with finite thickness volume which lies outside of the domain. We introduce a function space on the volumetric boundary region that serves as a trace space for these nonlocal problems and study the related extension results. Moreover, we discuss the consistency of the new nonlocal trace space with the classical W1−1/p,p(∂Ω) space as the size of nonlocal interaction tends to zero. In making this connection, we conduct an investigation on the relations between nonlocal interactions on a larger domain and the induced interactions on its subdomain. The various forms of trace, embedding and extension theorems may then be viewed as consequences in different scaling limits.