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 $u\mapsto (u,{\partial }_{n}u,\dots ,{\partial }_n^{k}u){|}_{\partial K}$. 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.

51 pages, 1 figure