Stable Liftings of Polynomial Traces on Tetrahedra

Abstract

On the reference tetrahedron \(K\), we construct, for each \(k \in {\mathbb {N}}_0\), a right inverse for the trace operator \(u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^k u)|_{\partial K}\). The operator is stable as a mapping from the trace space of \(W^{s, p}(K)\) to \(W^{s, p}(K)\) for all \(p \in (1, \infty )\) and \(s \in (k+1/p, \infty )\). Moreover, if the data is the trace of a polynomial of degree \(N \in {\mathbb {N}}_0\), then the resulting lifting is a polynomial of degree N. One consequence of the analysis is a novel characterization for the range of the trace operator.