On Trace Theorems for Sobolev Spaces

We survey a few trace theorems for Sobolev spaces on $N$-dimensional Euclidean domains. We include known results on linear subspaces, in particular hyperspaces, and smooth boundaries, as well as less known results for Lipschitz boundaries, including Besov’s Theorem and other characterizations of traces on planar domains, polygons in particular, in the spirit of the work of P. Grisvard. Finally, we present a recent approach, originally developed by G. Auchmuty in the case of the Sobolev space $H^1(\mathrm{\Omega })$ on a Lipschitz domain $\mathrm{\Omega }$, and which we have further developed for the trace spaces of $H^k(\mathrm{\Omega })$, $k\ge 2$, by using Fourier expansions associated with the eigenfunctions of new multi-parameter polyharmonic Steklov problems.