Obstructions to the extension problem of Sobolev mappings

Abstract

Let $M$ and $N$ be compact manifolds with $\partial M\ne\emptyset$. We show that when $1< p< \dim M$, there are two different obstructions to extending a map in $W^{1-1/p,p}(\partial M,N)$ to a map in $W^{1,p}(M,N)$. We characterize one of these obstructions which is topological in nature. We also give properties of the other obstruction. For some cases, we give a characterization of $f\in W^{1-1/p,p}(\partial M,N)$ which has an extension $F\in W^{1,p}(M,N)$.