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Homogenization of Nonlinear Elliptic PDEs with Oscillating Data : 진동하는 데이타를 가진 타원형 비선형 방정식의 균질화 문제

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dc.contributor.advisor이기암-
dc.contributor.author유민하-
dc.date.accessioned2017-07-14T00:40:12Z-
dc.date.available2017-07-14T00:40:12Z-
dc.date.issued2013-08-
dc.identifier.other000000012946-
dc.identifier.urihttps://hdl.handle.net/10371/121265-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2013. 8. 이기암.-
dc.description.abstract이 학위논문에서 우리는 주로 비선형 편미분 방정식의 균질화 문제를 다루고 있다. 첫 번째로는 다공된 영역에서의 균질화 문제를 연구하였다. 우리는 흔히 소프트 인클루전이라고 불리우는 문제의 결과를 좀 더 일반적인 비선형이고 다이버젼스 타입이 아닌 방정식으로 확장하였다. 두 번째로는, Cioranescu와 Murat에 의해서 발견된 Strange Term Behavior에 대한 결과를 좀 더 일반적인 초평면으로 확장하였다. 마지막으로 우리는 경계에 진동하는 데이타가 주어져 있을 경우의 균질화 문제를 연구하였다. 우리는 각 입실론 문제의 해의 수렴성에 관한 결과를 얻었다.-
dc.description.abstractIn this dissertation, we study three homogenization problems. First, we consider the homogenization in a perforated domain. We develop the viscosity method to get a homogenization result of semi linear PDE of non-divergence form in a perforated domain. Secondly, we study high oscillating problems. We extend Cioranescu and Murats result, [CM], to general hyper-planes. Finally, we consider the homogenization problem with oscillating boundary data. We get the local uniform convergence of
the solutions of each epsion problems.		-
dc.description.tableofcontentsAbstract i
1 Introduction 1
1.1 What is the Homogenization? 1
1.2 A preview of thesis 3
2 Preliminaries 6
2.1 Existence of Viscosity Solutions 6
2.2 Some Regularity Results of Viscosity Solutions 10
3 Homogenization of the Soft Inclusion 13
3.1 Introduction 13
3.1.1 Soft inclusions 13
3.1.2 Main Theorems 16
3.2 Compatibility Condition 17
3.2.1 Existence and Regularity of Periodic Viscosity Solution 18
3.2.2 Existence and Uniqueness of gamma 23
3.2.3 Examples satisfying the compatibility condition 26
3.3 First Corrector 27
3.3.1 Existence and Regularity 27
3.4 Second Corrector and Uniform Ellipticity of Effective Equation 31
3.4.1 Existence of the Second Corrector and the Effective Equation 32
3.4.2 Uniformly Ellipticity and Continuity of Effective Equation 35
3.5 Homogenization 39
3.5.1 Proof of Theorem 3.1.2 39
3.5.2 Construction of Barriers When the Domain is Convex 44
3.5.3 Construction of barriers for the non-convex domain 47
3.6 Gradient Estimate 51
3.6.1 Discrete Gradient Estimate 51
3.6.2 Epsilon-Flatness and global epsilon Lipschitz estimate 53
4 Highly Oscillating Thin Obstacles 56
4.1 Introduction 56
4.1.1 Formulation of the problem 56
4.1.2 Related Works 57
4.1.3 Main Theorem 58
4.1.4 Heuristic arguments and computation of the critical rate 62
4.2 Correctors 67
4.3 Proof of Theorem 4.1.1 79
4.4 The Case of General Hyper-Surfaces 81
4.4.1 Effective equations 83
4.5 Uniform distribution mod 1 85
4.5.1 Known Result for the 1-dimensional sequence 86
4.5.2 Application to the sequence in R^n 89
5 Homogenization of Oscillating Boundary Data 96
5.1 Introduction 96
5.2 Functions Defined on a Half-plain 99
5.3 Correctors 106
5.4 The Continuity of g 116
5.5 The Effective Solution 120
5.6 Proof of Theorem 5.1.1 129
Abstract (in Korean) 139		-
dc.formatapplication/pdf-
dc.format.extent2018125 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subjectHomogenization-
dc.subjectPerforated domain-
dc.subjectObstacle Problem-
dc.subjectOscillating boundary data-
dc.subjectnon-divergence elliptic equation-
dc.subject.ddc510-
dc.titleHomogenization of Nonlinear Elliptic PDEs with Oscillating Data-
dc.title.alternative진동하는 데이타를 가진 타원형 비선형 방정식의 균질화 문제-
dc.typeThesis-
dc.description.degreeDoctor-
dc.citation.pagesiii, 138-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2013-08-
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