Characterization of the traces on the boundary of functions in magnetic Sobolev spaces

Abstract

We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field $A$ is differentiable and its exterior derivative corresponding to the magnetic field $dA$ is bounded. In particular, we prove that, for $d\ge 1$ and $p>1$, the trace of the magnetic Sobolev space $W_A^{1,p}({\mathbb{R}}_{+}^{d+1})$ is exactly $W_{A^{\parallel }}^{1-1/p,p}({\mathbb{R}}^d)$ where $A^{\parallel }(x)=(A_1,\dots ,A_d)(x,0)$ for $x\in {\mathbb{R}}^d$ with the convention $A=(A_1,\dots ,A_{d+1})$ when $A\in C^1(\overline{{\mathbb{R}}_{+}^{d+1}},{\mathbb{R}}^{d+1})$. We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space.

24 pages