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Analysis and PDE on metric measure s...

Zhou, Xiaodan.

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Analysis and PDE on metric measure spaces: Sobolev functions and viscosity solutions.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Analysis and PDE on metric measure spaces: Sobolev functions and viscosity solutions./
作者:	Zhou, Xiaodan.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2016,
面頁冊數:	150 p.
附註:	Source: Dissertation Abstracts International, Volume: 78-05(E), Section: B.
Contained By:	Dissertation Abstracts International78-05B(E).
標題:	Theoretical mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=10298891
ISBN:	9781369419061

Analysis and PDE on metric measure spaces: Sobolev functions and viscosity solutions.
Zhou, Xiaodan.

Analysis and PDE on metric measure spaces: Sobolev functions and viscosity solutions. - Ann Arbor : ProQuest Dissertations & Theses, 2016 - 150 p.

Source: Dissertation Abstracts International, Volume: 78-05(E), Section: B.

Thesis (Ph.D.)--University of Pittsburgh, 2016.

We study analysis and partial differential equations on metric measure spaces by investigating the properties of Sobolev functions or Sobolev mappings and studying the viscosity solutions to some partial differential equations.

ISBN: 9781369419061Subjects--Topical Terms:

3173530
Theoretical mathematics.

Analysis and PDE on metric measure spaces: Sobolev functions and viscosity solutions.
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245 1 0 $a Analysis and PDE on metric measure spaces: Sobolev functions and viscosity solutions.
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300 $a 150 p.
500 $a Source: Dissertation Abstracts International, Volume: 78-05(E), Section: B.
500 $a Advisers: Piotr Hajlasz; Juan J. Manfredi.
502 $a Thesis (Ph.D.)--University of Pittsburgh, 2016.
520 $a We study analysis and partial differential equations on metric measure spaces by investigating the properties of Sobolev functions or Sobolev mappings and studying the viscosity solutions to some partial differential equations.
520 $a 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 construct a large class of pathological n-dimensional spheres in Rn+1 by showing that for any Cantor set C ⊂ Rn+1 there is a topological embedding f : Sn → Rn+1 of the Sobolev class W1,n whose image contains the Cantor set C.
520 $a 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 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 ow equations and normalized p-Laplace equations in the Euclidean space.
590 $a School code: 0178.
650 4 $a Theoretical mathematics. $3 3173530
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710 2 $a University of Pittsburgh. $3 958527
773 0 $t Dissertation Abstracts International $g 78-05B(E).
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792 $a 2016
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