Trace Theorems for some Nonlocal Function Spaces with Heterogeneous Localization

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. In this work, we present stronger and more general versions of such a trace theorem in a new nonlocal function space ${\mathcal{S}}(\Omega)$ satisfying $H^1(\Omega)\subset {\mathcal{S}}(\Omega)\subset L^2(\Omega)$. The new space ${\mathcal{S}}(\Omega)$ is associated with a nonlocal norm characterized by a nonlocal interaction kernel that is defined heterogeneously with a special localization feature on the boundary. Through the heterogeneous localization, we are able to show that the $H^{1/2}$ norm of the trace on the boundary can be controlled by the nonlocal norm that is weaker than the classical $H^1$ norm. In fact, the trace theorems can be essentially shown without imposing any extra regularity of the function in the interior of the domain other than being square integrable. Implications of the new trace theorems for the coupling of local and nonlocal e...

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