### Obstructions to the extension problem of Sobolev mappings

DOI: http://dx.doi.org/10.12775/TMNA.2003.021

#### 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 $F\in W^{1,p}(M,N)$.

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 $F\in W^{1,p}(M,N)$.

#### Keywords

Sobolev mappings; extension problem; trace spaces; obstruction theory

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