Nonlocal trace spaces and extension results for nonlocal calculus
Nonlocal trace spaces and extension results for nonlocal calculus
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非局部微积分的非局部迹空间和扩展结果
DOI：
10.1016/j.jfa.2022.109453
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发表时间：
2022
影响因子：
1.7
通讯作者：
Yu, Yue
中科院分区：
文献类型：
--
作者：
Du, Qiang;Tian, Xiaochuan;Wright, Cory;Yu, Yue
对于给定的Lipschitz域Ω，W1，p(Ω)的迹空间是W1−1/p，p(∂Ω)是经典结果，即任何W1，p(Ω)函数在其余维-1边界−上都有一个明确定义的W1∂Ω1/p，p(Ω)迹，并且任何W1−1/p，p(∂Ω)函数都可以扩展为W1，p(∂Ω)函数.在这项工作中，我们研究了具有有限范围非局部相互作用的积分微分算子的非局部Dirichlet问题的函数空间，并给出了它们的迹空间的刻画。对于这些非局部Dirichlet问题，边界条件通常施加在位于区域外的有限厚度体积的区域上。我们在体积边界区域上引入一个函数空间，作为这些非局部问题的迹空间，并研究了相关的推广结果。此外，我们还讨论了当非局部相互作用的规模趋于零时，新的非局部迹空间与经典的W1−1/p，p(∂Ω)空间的相合性。为了建立这种联系，我们研究了较大区域上的非局域相互作用与其子域上的诱导相互作用之间的关系。然后，各种形式的迹、嵌入和扩张定理可以被视为不同尺度限制下的结果。
For a given Lipschitz domain Ω, it is a classical result that the trace space of W 1, p (Ω) is W 1− 1/p, p (∂ Ω), namely any W 1, p (Ω) function has a well-defined W 1− 1/p, p (∂ Ω) trace on its codimension-1 boundary∂ Ω and any W 1− 1/p, p (∂ Ω) function on∂ Ω can be extended to a W 1, 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 W 1− 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.