Heterogeneously localized nonlocal operators, boundary traces and variational problems

Qiang Du Columbia University Xiaochuan Tian Columbia University

Analysis of PDEs mathscidoc:1703.03010

Proceedings of ICCM 2016, 2017
It is a classical result of Sobolev spaces that any $H^1$ function has a well-defined $H^{1/2}$ trace on the boundary of a sufficient regular domain. We discuss its recent extensions given in \cite{TiDu16} in some heterogeneously localized nonlocal function spaces. The new trace theorems are stronger and more general than the classical result. They can be established essentially for all functions having only square integrability away from the boundary or in any compact subset of interior domain. Yet, the heterogeneous localization offers the necessary regularity precisely at the boundary to have well-defined traces. A consequence is that we may study associated Dirichlet type boundary value problems, as well as the coupling of local and nonlocal equations through co-dimension-1 interfaces. %Their derivations not only involve various extensions of classical inequalities but also %require new techniques absent from traditional approaches.
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  title={Heterogeneously localized nonlocal operators, boundary traces and variational problems},
  author={Qiang Du, and Xiaochuan Tian},
  booktitle={Proceedings of ICCM 2016},
Qiang Du, and Xiaochuan Tian. Heterogeneously localized nonlocal operators, boundary traces and variational problems. 2017. In Proceedings of ICCM 2016. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170314193005127172701.
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