On extension of functions with zero trace on a part of the boundary (Q2367787)

An extension operator \(\Pi: W^{m,p}(\Omega)\to W^{m,p}(\mathbb{R}^ n)\) is constructed which allows to control the propagation of the support of the extended function. In particular, it is shown that if \(u\in W^{m,p}(\Omega)\) has zero trace on a part \(M\) of the boundary \(\partial\Omega\) and if there is a finite cone \(C\) such that \((x+ C)\cap\Omega=\emptyset\) for all \(x\in M\), then the extension operator satisfies -- under certain regularity assumptions on the boundary of \(M\) -- \[ (\text{supp }\Pi u)\cap (x+ C)= \emptyset \qquad\text{for all } x\in M. \] The operator \(\Pi\) is constructed by more or less standard methods via localization and flattening.