Timeline for Can a $W^{1,2}$ map from the disk to the circle restrict to a degree one map on the boundary?
Current License: CC BY-SA 3.0
9 events
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| Feb 17, 2017 at 4:07 | history | edited | Yasha Berchenko-Kogan | CC BY-SA 3.0 |
added a reference
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| Feb 17, 2017 at 3:57 | history | edited | Yasha Berchenko-Kogan | CC BY-SA 3.0 |
added 53 characters in body
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| Feb 15, 2017 at 20:40 | history | edited | Yasha Berchenko-Kogan | CC BY-SA 3.0 |
corrected the formula by adding a factor of 1/6.
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| Feb 14, 2017 at 21:46 | history | edited | Yasha Berchenko-Kogan | CC BY-SA 3.0 |
added the higher dimensional generalization.
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| Feb 14, 2017 at 21:40 | history | edited | Yasha Berchenko-Kogan | CC BY-SA 3.0 |
added the higher dimensional generalization.
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| Feb 14, 2017 at 20:54 | history | edited | Yasha Berchenko-Kogan | CC BY-SA 3.0 |
replaced the proof that circle-valued smooth functions are dense in the Sobolev space with a citation.
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| Feb 14, 2017 at 20:43 | comment | added | Yasha Berchenko-Kogan | Tom Mrowka pointed me to a Schoen and Uhlenbeck paper that discusses this density issue. See section 4. There's also a recent paper whose introduction seems to provide a good overview of what's currently known about smooth approximation for general Sobolev maps between manifolds. In particular, smooth maps $C^\infty(M;N)$ between compact manifolds are indeed dense in the borderline Sobolev spaces $W^{1,n}(M;N)$, where $n=\dim M$. I'll update my answer to reflect this. | |
| Feb 11, 2017 at 17:38 | comment | added | Tim Carson | The point about non-density of $C^{\infty}(D^2, S^1)$ in $W^{1,p}(D^2;S^1)$ is interesting. It seems that maybe the $C^{\infty}$ functions are dense in $W^{1,2}$, because of this degree argument. Intuitively, start with an approximating sequence $g_n$ of $C^{\infty}(D^2; \mathbb{C})$ functions and make them $S^1$ valued outside a disk of radius $(1/n)$. By the continuity of the integral you observed (applied to the small disk) you may argue that $g_n$ can be homotopic to a constant map in the small disk, and so they can be $S^1$ valued in the small disk as well. | |
| Feb 11, 2017 at 17:05 | history | answered | Yasha Berchenko-Kogan | CC BY-SA 3.0 |