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α-Gauss Curvature Flows and Free Boundary Problems

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dc.contributor.advisor이기암-
dc.contributor.author김라미-
dc.date.accessioned2017-07-14T00:40:17Z-
dc.date.available2017-07-14T00:40:17Z-
dc.date.issued2013-08-
dc.identifier.other000000013006-
dc.identifier.urihttps://hdl.handle.net/10371/121267-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 8. 이기암.-
dc.description.abstract본 학위논문은 알파 지수 가우스 곡률을 속력으로 갖는 n+1차원 유클리드 공간에 있는 n차원 순볼록 초곡면의 변형을 연구한다. 알파의 범위가 1/n-
dc.description.tableofcontentsAbstract i
1 Introduction ....... 1
2 Preliminaries ........... 6
2.1 Definitions and terminology ................... 6
2.1.1 Metric, the second fundamental form and curvature . . 6
2.1.2 Support function ..................... 7
2.2 Evolutions of the geometric quantities. . . . . . . . . . . . . . 8
2.2.1 Evolutions of metric, the second fundamental form, and curvature....................... 8
2.2.2 Evolutions with respect to the standard metric g_ij on S^n ..9
3 α-Gauss Curvature Flows of an n-Dimensional Compact Strictly Convex Hypersurface .......... 11
3.1 Main theorem........................... 11
3.2 Curvature estimates ....................... 12
3.3 Integral quantity and asymptotic behavior of hypersurface. .15
3.4 Existence of solutions and proof of main theorem . . . . . . . 32
3.4.1 Short time existence ................... 32
3.4.2 Long time existence.................... 32
4 α-Gauss Curvature Flows with Flat Sides and Free Boundary Problems 34
4.1 Preliminaries ........................... 34
4.1.1 The balance of terms................... 34
4.1.2 Conditions for f...................... 35
4.1.3 The concept of regularity................. 36
4.2 Main theorems .......................... 38
4.3 Convex surfaces.......................... 39
4.3.1 Curvature estimates ................... 39
4.3.2 Strict convexity away from the flat spot . . . . . . . . 41
4.3.3 Proof of Theorem 4.2.1.................. 43
4.3.4 A waiting time effect ................... 43
4.4 Optimal gradient estimate near free boundary . . . . . . . . . 44
4.4.1 Finite and non-degenerate speed of level sets . . . . . . 44
4.4.2 Gradient estimates .................... 45
4.5 Second derivative estimates ................... 46
4.5.1 Decay rate of α-Gauss curvature . . . . . . . . . . . . 47
4.5.2 Upper bound of the curvature of level sets . . . . . . . 47
4.5.3 Aronson-Benilan type estimate . . . . . . . . . . . . . 52
4.5.4 Global optimal regularity................. 54
4.5.5 Decay rates of second derivatives . . . . . . . . . . . . 57
4.6 Higher regularity ......................... 58
4.6.1 Local change of coordinates ............... 58
4.6.2 Class of linearized equation ............... 58
4.6.3 Regularity theory..................... 61
Abstract (in Korean) .............. 67		-
dc.formatapplication/pdf-
dc.format.extent1601531 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subject가우스 곡률 흐름-
dc.subject알파 가우스 곡률 흐름의 정칙성-
dc.subject자유 경계 문제-
dc.subject비선형 포물형 편미분 방정식-
dc.subject.ddc510-
dc.titleα-Gauss Curvature Flows and Free Boundary Problems-
dc.typeThesis-
dc.description.degreeDoctor-
dc.citation.pages1, 66-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2013-08-
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