장애물문제의 정칙성

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2019. 2. 이기암.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.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^+)^2.1 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 . 80Docto

Properties of obstacle problem and free boundary problem

학위논문 (석사)-- 서울대학교 대학원 : 수리과학부, 2014. 8. 이기암.This paper is a paper which is written based on the contents of [1] and introduction of obstacle problem for nonlinear second-order parabolic operator. In chapter 1, we introduce classical obstacle problem and we deal with existence, uniqueness and C11 regularity of solution of the problem. In chapter 2, we show C11 regularity of solution of Obstacle-type problem. In chapter 3, we prove some elementary properties of free boundary. In chapter 4, We reference [2] to show the continuity of solution of obstacle problem for nonlinear second-order parabolic operator. Key1 The classical obstacle problem 1 1.1 The obstacle problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Existense and uniqueness of the solution of the obstacle problems . . . . . . . 4 1.3 W2p regularity of the solution of the classical obstacle problem . . . . . . . . . 6 1.4 C11 regularity of the solution of the classical obstacle problem . . . . . . . . . 8 2 Optimal regularity of solutions of obstacle problems 10 2.1 Model problems ABC and OT1 ?? OT2 . . . . . . . . . . . . . . . . . . . . . 10 2.2 ACF monotonicity formula and generalizations . . . . . . . . . . . . . . . . . 11 2.3 Optimal regularity in OT1 ?? OT2 . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Preliminary analysis of the free boundary 20 3.1 Nondegeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.2 Lebesgue and Hausdo measures of the free boundary . . . . . . . . . . . . . 22 3.3 Classes of solutions, rescalings, and blowups . . . . . . . . . . . . . . . . . . 26 4 Obstacle problem for nonlinear second-order parabolic operator 29 4.1 Viscosity solution of parabolic equations . . . . . . . . . . . . . . . . . . . . . 29 4.2 The existence and the continuity theory . . . . . . . . . . . . . . . . . . . . . 30Maste

Lanthanum Strontium Cobalt Ferrite Oxide Series electrode material for Solid Oxide fuel cell and Manufacturing Method thereof

본 발명은 고체산화물 연료전지에 관한 것으로, 스트론튬, 코발트 조성 변화를 통해 산소 이온 전도성 및 전자 전도성이 향상된 고체산화물 연료전지의 공기극용 나노 입자 LSCF 계 고체산화물 연료전지 전극 소재 및 그 제조 방법에 관한 것으로서, 상기 LSCF 계 고체산화물 연료전지 전극 소재는, La1-xSrxCo1-yFeyO3-δ로 표현되며, 0.78 ≤ x ≤ 0.82이고, 0.18 ≤ y ≤ 0.22 이며, 0 ≤ δ ≤ 1인 란탄 스트론튬 코발트 철 복합산화물인 것을 특징으로 한다