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Register a new userFractional Hardy-type and trace theorems for nonlocal function spaces with heterogeneous localization
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In this work, we prove some trace theorems for function spaces with a nonlocal character that contain the classical $W^{s,p}$ space as a subspace. The result we obtain generalizes well known trace theorems for $W^{s,p}(Omega)$ functions which has a well-defined $W^{1-s/p, p}$ trace on the boundary of a domain with sufficient regularity. The new generalized spaces are associated with norms that are characterized by nonlocal interaction kernels defined heterogeneously with a special localization feature on the boundary. Intuitively speaking, the class contains functions that are as rough as an $L^p$ function inside the domain of definition but as smooth as a $W^{s,p}$ function near the boundary. Our result is that the $W^{1-s/p, p}$ norm of the trace on the boundary such functions is controlled by the nonlocal norms that are weaker than the classical $W^{s, p}$ norm. These results are improvement and refinement of the classical results since the boundary trace can be attained without imposing regularity of the function in the interior of the domain. They also extend earlier results in the case of $p=2$. In the meantime, we prove Hardy-type inequalities for functions in the new generalized spaces that vanish on the boundary, showing them having the same decay rate to the boundary as functions in the smaller space $W^{s,p}(Omega)$. A Poincare-type inequality is also derived. An application of the new theory leads to the well-posedness of a nonlinear variational problem that allows possible singular behavior in the interior with imposed smoother data on the boundary.
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For a given Lipschitz domain $Omega$, it is a classical result that the trace space of $W^{1,p}(Omega)$ is $W^{1-1/p,p}(partialOmega)$, namely any $W^{1,p}(Omega)$ function has a well-defined $W^{1-1/p,p}(partialOmega)$ trace on its codimension-1 boundary $partialOmega$ and any $W^{1-1/p,p}(partialOmega)$ function on $partialOmega$ can be extended to a $W^{1,p}(Omega)$ function. Recently, Dyda and Kassmann (2019) characterize the trace space for nonlocal Dirichlet problems involving integrodifferential operators with infinite interaction ranges, where the boundary datum is provided on the whole complement of the given domain $mathbb{R}^dbackslashOmega$. In this work, we study function spaces for nonlocal Dirichlet problems with a finite range of nonlocal interactions, which naturally serves a bridging role between the classical local PDE problem and the nonlocal problem with infinite interaction ranges. 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 $W^{1-1/p,p}(partialOmega)$ 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.
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