Approximation in Sobolev Spaces between two manifolds and homotopy groups
We consider two compact manifolds M n and N k and the Sobolev spaces W 1,p (M n , N k ),for 1 < p < n = dim M n . We give a necessary and sufficient condition for smooth maps between M n and N k to be dense in W 1,p (M n , N k ). This condition can be simply stated in terms of homotopy groups, and is π[p](N k )= 0. In cases where such a condition does not hold, we show that we can approximate maps in W 1,p (M n , N k ) by maps smooth except on a singular set which has a simple shape. We consider also the problem of the weak density of smooth maps.
KeywordsFinite Number Sobolev Space Point Singularity Weak Topology Homotopy Group
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