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

Nguyen, Hoai-Minh Van Schaftingen, Jean [UCL]

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≥1 and p>1, the trace of the magnetic Sobolev space W1,pA(Rd+1+) is exactly W1−1/p,pA∥(Rd) where A∥(x)=(A1,…,Ad)(x,0) for x∈Rd with the convention A=(A1,…,Ad+1) when A∈C1(Rd+1+,Rd+1). We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space.