Zhou, Xiaodan
(2016)
Analysis and PDE on metric measure spaces: Sobolev functions and Viscosity solutions.
Doctoral Dissertation, University of Pittsburgh.
(Unpublished)
Abstract
We study analysis and partial differential equations on metric measure spaces by investigating the property of Sobolev functions or Sobolev mappings and studying the viscosity solutions to some partial differential equations.
This manuscript consists of two parts. The first part is focused on the theory of Sobolev spaces on metric measure spaces. We investigate the continuity of Sobolev functions in the critical case in some general metric spaces including compact connected one-dimensional spaces and fractals. We also constructe a large class of pathological $n$-dimensional spheres in $\mathbb{R}^{n+1}$ by showing that for any Cantor set $C\subset\mathbb{R}^{n+1}$ there is a topological embedding
$f:\mathbb{S}^n\to\mathbb{R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.
The second part is focused on the theory of viscosity solutions for nonlinear partial differential equations in metric spaces, including the Heisenberg group as an important special case. We study Hamilton-Jacobi equations on the Heisenberg group $\mathbb{H}$ and show uniqueness of viscosity solutions with exponential growth at infinity. Lipschitz and horizontal convexity preserving properties of these equations under appropriate assumptions are also investigated. In this part, we also study a recent game-theoretic approach to the viscosity solutions of various equations, including deterministic and stochastic games. Based on this interpretation, we give new proofs of convexity preserving properties of the mean curvature flow equations and normalized $p$-Laplace equations in the Euclidean space.
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Details
| Item Type: |
University of Pittsburgh ETD
|
| Status: |
Unpublished |
| Creators/Authors: |
|
| ETD Committee: |
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| Date: |
3 October 2016 |
| Date Type: |
Publication |
| Defense Date: |
16 June 2016 |
| Approval Date: |
3 October 2016 |
| Submission Date: |
10 August 2016 |
| Access Restriction: |
No restriction; Release the ETD for access worldwide immediately. |
| Number of Pages: |
149 |
| 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: |
Sobolev spaces, metric measure spaces, Heisenberg group, viscosity solution, convexity preserving, game-theoretic methods |
| Date Deposited: |
03 Oct 2016 22:20 |
| Last Modified: |
15 Nov 2016 14:35 |
| URI: |
http://d-scholarship.pitt.edu/id/eprint/29165 |
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