Liu, Zhuomin
(2013)
Conformal mappings and isometric immersions under second order Sobolev regularity.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
We consider two classes of vector valued functions with conformal constraint- conformal mappings from an $n$-dimensional domain into $\bbbr^n$ and isometric immersions of an $n$-dimensional domain into $\bbbr^{n+1}$ (co-dimension one) for $n\geq 3$.
Iwaniec and Martin proved that in even dimensions $n\geq 3$, $W_{\rm{loc}}^{1,n/2}$ conformal mappings are M\"{o}bius transformations and they conjectured that it should also be true in odd dimensions. In the first part of this manuscript, we prove this theorem for a conformal map $f\in W_{\rm{loc}}^{1,1}$ in dimension $n\geq 3$ under one additional assumption that the norm of the first order derivative $|Df|$ satisfies $|Df|^p\in W_{\rm{loc}}^{1,2}$ for $p\geq (n-2)/4$. This is optimal in the sense that if $|Df|^p\in W_{\rm{loc}}^{1,2}$ for $p< (n-2)/4$, it may not be a M\"{o}bius transform. This result shows the necessity of the Sobolev exponent in the Iwaniec-Martin conjecture.
In the second part, we prove the developability and $C_{\rm loc}^{1,1/2}$ regularity of $W^{2,2}$
isometric immersions of $n$-dimensional domains into $\bbbr^{n+1}$ for $n\geq 3$. The result is sharp in the sense that $W^{1,p}, 1\leq p\leq \infty$ and $W^{2,p}, 1\leq p<2$ isometric immersions may not be developable. Based on this result, we also prove that if the domain is $C^1$ and convex, smooth isometric immersions are strongly dense in this space.
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| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
|
| Date: |
29 January 2013 |
| Date Type: |
Publication |
| Defense Date: |
30 November 2012 |
| Approval Date: |
29 January 2013 |
| Submission Date: |
6 December 2012 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
157 |
| Institution: |
University of Pittsburgh |
| Schools and Programs: |
Dietrich School of Arts and Sciences > Mathematics |
| Degree: |
PhD - Doctor of Philosophy |
| Thesis Type: |
Doctoral Dissertation |
| Refereed: |
Yes |
| Uncontrolled Keywords: |
Conformal mappings, isometric immersions, Sobolev Spaces. |
| Date Deposited: |
29 Jan 2013 21:39 |
| Last Modified: |
15 Nov 2016 14:08 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/16801 |
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