Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Nonlocal Trace Spaces and Extension Results for Nonlocal Calculus   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. 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
$\mathbb{R}^d\backslash\Omega$. 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
$W^{1-1/p,p}(\partial\Omega)$ 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.

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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: math-ph
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The categories of this paper are: econ
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The categories of this paper are: stat
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The categories of this paper are: q-alg
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The categories of this paper are: dg-ga
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: cond-mat