# Analysis and PDE on metric measure spaces: Sobolev functions and Viscosity solutions

Zhou, Xiaodan (2016) Analysis and PDE on metric measure spaces: Sobolev functions and Viscosity solutions. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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## 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Status: Unpublished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:
Zhou, Xiaodanbetty2178@gmail.comXIZ78
ETD Committee:
Committee ChairHajlasz, Piotrhajlasz@pitt.eduHAJLASZ
Committee CoChairManfredi, Juanmanfredi@pitt.eduMANFREDI
Committee MemberLennard, Christopherlennard@pitt.eduLENNARD
Committee MemberGaldi, Paologaldi@pitt.eduGALDI
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