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The Magnetic Ginzburg-Landau Equatio...

Halavati, Aria,

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The Magnetic Ginzburg-Landau Equation: Stability, Regularity and Rigidity /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	The Magnetic Ginzburg-Landau Equation: Stability, Regularity and Rigidity // Aria Halavati.
作者:	Halavati, Aria,
面頁冊數:	1 electronic resource (249 pages)
附註:	Source: Dissertations Abstracts International, Volume: 86-12, Section: B.
Contained By:	Dissertations Abstracts International86-12B.
標題:	Mathematics. -
電子資源:	https://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=31848673
ISBN:	9798286425242

The Magnetic Ginzburg-Landau Equation: Stability, Regularity and Rigidity /
Halavati, Aria,

The Magnetic Ginzburg-Landau Equation: Stability, Regularity and Rigidity /Aria Halavati. - 1 electronic resource (249 pages)

Source: Dissertations Abstracts International, Volume: 86-12, Section: B.

We study the magnetic Ginzburg-Landau energy in the critical coupling, also known as the abelian--Higgs model. It is known that entire solutions of the abelian-Higgs model blow down to (generalized) minimal submanifolds. We show that, in the so-called multiplicity one regime, critical points inherit an improvement of flatness and a rigidity property from their blow-down limit.This thesis consists of three parts. In the first two parts we develop the necessary toolbox for part three.First (in [44]), we develop a new class of weighted inequalities on any two-manifold (with boundary). Second (in [45]), using these inequalities and a selection principle (inspired by the quantitative isoperimetric inequality) we prove a sharp quantitative stability for the abelian-Higgs model in two dimensions.Third, (in a collaboration with Guido De Philippis and Alessandro Pigati in [28]) we leverage these tool to develop a large scale regularity theory for the zero set of solutions (in the spirit of Allard's). In fact, in the multiplicity one regime we show the uniqueness of blowdowns. Then we classify solutions in dimensions n 2.

English

ISBN: 9798286425242Subjects--Topical Terms:

515831
Mathematics.
Subjects--Index Terms:

Abelian-Higgs model

The Magnetic Ginzburg-Landau Equation: Stability, Regularity and Rigidity /
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520 $a We study the magnetic Ginzburg-Landau energy in the critical coupling, also known as the abelian--Higgs model. It is known that entire solutions of the abelian-Higgs model blow down to (generalized) minimal submanifolds. We show that, in the so-called multiplicity one regime, critical points inherit an improvement of flatness and a rigidity property from their blow-down limit.This thesis consists of three parts. In the first two parts we develop the necessary toolbox for part three.First (in [44]), we develop a new class of weighted inequalities on any two-manifold (with boundary). Second (in [45]), using these inequalities and a selection principle (inspired by the quantitative isoperimetric inequality) we prove a sharp quantitative stability for the abelian-Higgs model in two dimensions.Third, (in a collaboration with Guido De Philippis and Alessandro Pigati in [28]) we leverage these tool to develop a large scale regularity theory for the zero set of solutions (in the spirit of Allard's). In fact, in the multiplicity one regime we show the uniqueness of blowdowns. Then we classify solutions in dimensions n 2.
546 $a English
590 $a School code: 0146
650 4 $a Mathematics. $3 515831
650 4 $a Theoretical mathematics. $3 3173530
650 4 $a Applied mathematics. $3 2122814
653 $a Abelian-Higgs model
653 $a Calculus of variations
653 $a Geometric measure theory
653 $a Ginzburg-Landau energy
653 $a Partial differential equations
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690 $a 0642
690 $a 0364
710 2 $a New York University. $b Mathematics. $e degree granting institution. $3 3765635
720 1 $a De Philippis, Guido $e degree supervisor.
720 1 $a Lin, Fang-Hua $e degree supervisor.
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791 $a Ph.D.
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