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Eigenvalue Estimates, Minimal Hypers...

Tu, Yucheng.

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Eigenvalue Estimates, Minimal Hypersurfaces and Isoperimetric Inequalities.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Eigenvalue Estimates, Minimal Hypersurfaces and Isoperimetric Inequalities./
作者:	Tu, Yucheng.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2021,
面頁冊數:	84 p.
附註:	Source: Dissertations Abstracts International, Volume: 83-01, Section: B.
Contained By:	Dissertations Abstracts International83-01B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28542181
ISBN:	9798516948572

Eigenvalue Estimates, Minimal Hypersurfaces and Isoperimetric Inequalities.
Tu, Yucheng.

Eigenvalue Estimates, Minimal Hypersurfaces and Isoperimetric Inequalities. - Ann Arbor : ProQuest Dissertations & Theses, 2021 - 84 p.

Source: Dissertations Abstracts International, Volume: 83-01, Section: B.

Thesis (Ph.D.)--University of California, San Diego, 2021.

This item must not be sold to any third party vendors.

In this thesis we study three problems in the field of geometric analysis: eigenvalue estimate of non-linear operators, existence of minimal surfaces and isoperimetric problems. These problems are more or less related to the topic of geometric calculus of variations, which is the study of extreme points of functionals defined on manifolds.The first part is devoted to the study of lower bound of the principal eigenvalue of a family of non-linear elliptic operator Lp. Using a gradient and maximum comparison technique by Koerber together with ideas from Li and Wang, we proved that on a compact metric measure space(possibly with convex boundary) (M,g,m) with curvature-dimension condition BE (κ, N) (κ < 0), if L is a elliptic diffusion operator whose invariant measure is m, then the principal eigenvalue of Lp is bounded below by the first eigenvalue of a one-dimensional ODE with Neumann boundary condition. We showed that this is sharp result by constructing examples of metric measure space M on which the eigenvalue problem of Lp degenerates into the model equation problem. This work extends the κ = 0 case proved in Koerber.The second part is devoted to the study of existence of free boundary minimal hypersurfaces in compact manifolds, from a min-max theoretical point of view. Following the ideas from Ambrozio and Marquez, we prove that in a simply connected compact manifold (M, ∂M, g) under certain conditions) with its metric that is locally maximising the width of M, there is a sequence of equidistributed free boundary minimal hypersurfaces.The third part is devoted to the study of anisotropic isoperimetric inequality for regions outside of a ball in ℝn. Based on Alexandrov-Bakelman-Pucci's Method, we use the concept of generalized normal cone introduced by Liu, Wang, and Weng, to show that for any region outside a Euclidean ball, its isoperimetric ratio has a lower bound that can only be achieved by a half-Wulff shape cut by a half-space.

ISBN: 9798516948572Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Eigenvalue estimates

Eigenvalue Estimates, Minimal Hypersurfaces and Isoperimetric Inequalities.
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500 $a Source: Dissertations Abstracts International, Volume: 83-01, Section: B.
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502 $a Thesis (Ph.D.)--University of California, San Diego, 2021.
506 $a This item must not be sold to any third party vendors.
520 $a In this thesis we study three problems in the field of geometric analysis: eigenvalue estimate of non-linear operators, existence of minimal surfaces and isoperimetric problems. These problems are more or less related to the topic of geometric calculus of variations, which is the study of extreme points of functionals defined on manifolds.The first part is devoted to the study of lower bound of the principal eigenvalue of a family of non-linear elliptic operator Lp. Using a gradient and maximum comparison technique by Koerber together with ideas from Li and Wang, we proved that on a compact metric measure space(possibly with convex boundary) (M,g,m) with curvature-dimension condition BE (κ, N) (κ < 0), if L is a elliptic diffusion operator whose invariant measure is m, then the principal eigenvalue of Lp is bounded below by the first eigenvalue of a one-dimensional ODE with Neumann boundary condition. We showed that this is sharp result by constructing examples of metric measure space M on which the eigenvalue problem of Lp degenerates into the model equation problem. This work extends the κ = 0 case proved in Koerber.The second part is devoted to the study of existence of free boundary minimal hypersurfaces in compact manifolds, from a min-max theoretical point of view. Following the ideas from Ambrozio and Marquez, we prove that in a simply connected compact manifold (M, ∂M, g) under certain conditions) with its metric that is locally maximising the width of M, there is a sequence of equidistributed free boundary minimal hypersurfaces.The third part is devoted to the study of anisotropic isoperimetric inequality for regions outside of a ball in ℝn. Based on Alexandrov-Bakelman-Pucci's Method, we use the concept of generalized normal cone introduced by Liu, Wang, and Weng, to show that for any region outside a Euclidean ball, its isoperimetric ratio has a lower bound that can only be achieved by a half-Wulff shape cut by a half-space.
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