Nonlocal trace spaces and extension results for nonlocal calculus. (English) Zbl 1496.46031
Summary: For a given Lipschitz domain \(\Omega \), it is a classical result that the trace space of \(W^{1 , p}(\Omega)\) is \(W^{1 - 1 / p , p}(\partial \Omega)\), namely any \(W^{1 , p}(\Omega)\) function has a well-defined \(W^{1 - 1 / p , p}(\partial \Omega)\) trace on its codimension-1 boundary \(\partial \Omega\) and any \(W^{1 - 1 / p , p}(\partial \Omega)\) function on \(\partial \Omega\) can be extended to a \(W^{1 , p}(\Omega)\) 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 \(W^{1 - 1 / p , p}(\partial \Omega)\) 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.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 47G20 | Integro-differential operators |
| 35A23 | Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals |
| 35R11 | Fractional partial differential equations |
Keywords:
trace theorem; nonlocal function space; inverse trace theorem; finite nonlocal interactionsReferences:
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