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The Regularity of Obstacle Problems : 장애물문제의 정칙성
DC Field | Value | Language |
---|---|---|
dc.contributor.advisor | 이기암 | - |
dc.contributor.author | 박진완 | - |
dc.date.accessioned | 2019-05-07T07:01:06Z | - |
dc.date.available | 2019-05-07T07:01:06Z | - |
dc.date.issued | 2019-02 | - |
dc.identifier.other | 000000154972 | - |
dc.identifier.uri | https://hdl.handle.net/10371/152915 | - |
dc.description | 학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2019. 2. 이기암. | - |
dc.description.abstract | 이 박사학위 논문에서는 장애물 문제의 해의 정칙성과 자유경계의 정칙성을 다룬다. 특별히, 비 볼록 완전 비선형연산자의 장애물문제의 자유경계의 정칙성과 이중장애물 문제의 해의 정칙성과 자유경계의 정칙성을 다룬다.
비 볼록 완전 비선형연산자의 장애물문제의 자유경계의 정칙성을 증명하기 위해서, F(D^2u)=0의 해의 내부 C^{2,\alpha} 정칙성을 보였다. 라플라시안의 이중장애물 문제에서는 ACF 단조공식과 바이스 단조공식을 이용하였다. 이 단조공식은 완전 비선형연산자의 이중 장애물 문제에는 적용 될 수 없다. 그래서, 반공간 함수 \psi=c(x_n^+)^2를 위 장애물로 갖는 대역해 u에 대해 e가 e_n과 수직인 방향일 때, \partial_e u/x_n이 유한하다는 것을 이용하였다. | - |
dc.description.abstract | In this dissertation, we consider the regularity of solutions and the regularity of the free boundary of the obstacle problems. Specifically, we study the regularity of the free boundary of a non-convex fully nonlinear operator and the regularity of solutions and the free boundary of the double obstacle problem.
In order to prove the regularity of the free boundary of a non-convex fully nonlinear operator, we have the interior C^{2,\alpha} regularity of the solution of the Dirichlet problem for the non-convex fully nonlinear operator. In the double obstacle problem for Laplacian, we use the ACF monotonicity formula and the Weiss' monotonicity formula. The monotonicity formulas are not applicable for the double obstacle problem for fully nonlinear operator. Hence, we exploit the fact that the term \partial_e u/x_n is finite, where e is a direction orthogonal to e_n, for the global solution u with the half space function type upper obstacle \psi=c(x_n^+)^2. | - |
dc.description.tableofcontents | 1 Introduction 1
1.1 Introduction of Obstacle Problems . . . . . . . . . . . . . . . 1 1.2 A Preview of Dissertation . . . . . . . . . . . . . . . . . . . . 2 2 Preliminaries 5 2.1 Fully Nonlinear Operator . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Viscosity Solutions . . . . . . . . . . . . . . . . . . . 5 2.1.2 Regularity of the Solution of the Fully Nonlinear Operator 6 2.2 Rescaling, Blowup and Thickness assumption. . . . . . . . . . 8 3 Obstacle Problem for a Non-convex Fully Nonlinear Operator 10 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.2 The Conditions on Fully Nonlinear Operator and Level Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.1.3 Definitions . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.4 Main Theorems . . . . . . . . . . . . . . . . . . . . . 16 3.2 C^{2,\alpha} Regularity of Solutions for F(D^2u) = 0 . . . . . 18 3.3 Regularity of the Free Boundary . . . . . . . . . . . . . . . . 23 3.3.1 General Properties . . . . . . . . . . . . . . . . . . . 23 3.3.2 Convexity of Global Solutions u\in P_\infty(M) . . . . . . . 26 3.3.3 Directional Monotonicity . . . . . . . . . . . . . . . . 37 3.3.4 Proof of Theorem 3.1.1 and Corollary 3.1.2 . . . . . . 40 4 Double Obstacle Problem (Linear Case) 42 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 44 4.1.3 Main Results . . . . . . . . . . . . . . . . . . . . . . 46 4.2 Standard Results . . . . . . . . . . . . . . . . . . . . . . . . 46 4.2.1 Optimal regularity . . . . . . . . . . . . . . . . . . . 46 4.2.2 Non-degeneracy . . . . . . . . . . . . . . . . . . . . 47 4.3 Properties of Global Solutions . . . . . . . . . . . . . . . . . 49 4.3.1 Dimensionality Reduction and Positivity of Global Solutions with the Upper Obstacle \psi = \frac{a}{2}(x^+_1)^2 . . . . . . 49 4.3.2 Homogeneity of Blowup and Shrink-down of Global Solutions with the Upper Obstacle \psi = \frac{a}{2}(x^+_1)^2 . . . . . 54 4.4 Directional Monotonicity . . . . . . . . . . . . . . . . . . . . 57 4.5 Classification of Blowups . . . . . . . . . . . . . . . . . . . . 61 4.6 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . 62 5 Double Obstacle Problem (Fully Nonlinear Case) 65 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.1.1 Reduction of (FB) . . . . . . . . . . . . . . . . . . . 68 5.1.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1.3 Conditions on F = F(M, x) . . . . . . . . . . . . . . 69 5.1.4 Definitions . . . . . . . . . . . . . . . . . . . . . . . 69 5.1.5 Main Theorems . . . . . . . . . . . . . . . . . . . . . 70 5.2 Existence, Uniqueness and Optimal Regularity . . . . . . . . 72 5.2.1 Existence, uniqueness of W^{2,p} solution . . . . . . . . . 72 5.2.2 Optimal Regularity . . . . . . . . . . . . . . . . . . . 74 5.3 Regularity of the Free Boundary . . . . . . . . . . . . . . . . 77 5.3.1 Non-degeneracy . . . . . . . . . . . . . . . . . . . . 77 5.3.2 Classification of Global Solutions . . . . . . . . . . . 78 5.3.3 Directional Monotonicity and proof of Theorem 5.1.2 . 80 | - |
dc.language.iso | eng | - |
dc.publisher | 서울대학교 대학원 | - |
dc.subject.ddc | 510 | - |
dc.title | The Regularity of Obstacle Problems | - |
dc.title.alternative | 장애물문제의 정칙성 | - |
dc.type | Thesis | - |
dc.type | Dissertation | - |
dc.contributor.AlternativeAuthor | Jinwan Park | - |
dc.description.degree | Doctor | - |
dc.contributor.affiliation | 자연과학대학 수리과학부 | - |
dc.date.awarded | 2019-02 | - |
dc.contributor.major | 편미분방정식 | - |
dc.identifier.uci | I804:11032-000000154972 | - |
dc.identifier.holdings | 000000000026▲000000000039▲000000154972▲ | - |
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