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Regularity results for generalized elliptic problems in bounded domains
유계영역에서 정의된 일반화된 타원형 방정식의 정칙성

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dc.contributor.advisor변순식-
dc.contributor.author소형석-
dc.date.accessioned2017-07-14T00:43:00Z-
dc.date.available2017-07-14T00:43:00Z-
dc.date.issued2017-02-
dc.identifier.other000000141338-
dc.identifier.urihttps://hdl.handle.net/10371/121322-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2017. 2. 변순식.-
dc.description.abstractThree different types of problems will be studied in this thesis. The three problems are the G-Laplace equation in a convex domain, a quasilinear equation with p-growth condition in a quasiconvex domain and generalized steady Stokes system in a Reifenberg flat domain. In each problem, we focus on a gradient estimate of a weak solution.
At first, we prove local boundedness of of the gradient for the homogeneous G-Laplace equation in a convex domain under vanishing Neumann boundary condition. G is a Young function which is a non-decreasing convex
function such that G(0) = 0 and lim_{t→+∞}G(t)/t = +∞. In this problem, one of our interests is a convex domain, since Lipschitz regularity of a solution to even the Laplace equation cannot be obtained in a Lipschitz domain.
Next, we derive Calderón-Zygmund type estimate for the solution to quasilinear equation with p-growth condition in a quasiconvex domain, which is locally trapped by two convex domains. As far as the domain is concerned,
our regularity assumption on the boundary is weaker than any other one reported in this direction. In addition, we extend our result in Lebesgue spaces to Orlicz spaces.
In last chapter, We prove the global weighted L^q-estimates for the gradient of the weak solution and an associated pressure under the assumptions that the coefficients have small BMO (bounded mean oscillation) semi-norms and the domain is sufficiently flat in the Reifenberg sense. On the other hand, a given weight is assumed to belong to a Muckenhoupt class. Our result generalizes the global W 1,q estimate for a solution with respect to the Lebesgue measure for the Stokes system in a Lipschitz domain.
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dc.description.tableofcontents1 Introduction 1
2 Local Lipschitz regularity in convex domains 5
2.1 Overview 5
2.2 Elemental definitions and auxiliary results 7
2.2.1 Young function G and auxiliary results 7
2.3 Proof of the main theorem 11
3 Gradient estimates for elliptic equations in quasiconvex domains 23
3.1 Overview 23
3.2 Notation and preliminary results 26
3.3 Quasiconvex Domains 29
3.4 Proof of the Main Theorem 32
3.5 Gradient estimate in Orlicz spaces 60
4 Stokes system 64
4.1 Overview 64
4.2 Preliminaries 68
4.2.1 Notations 68
4.2.2 Weighted Lebesgue spaces and technical lemmas 68
4.2.3 Existence and energy estimates of weak solution pairs 71
4.3 Gradient estimates in L^q_ω 74
Abstract (in Korean) 101
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dc.formatapplication/pdf-
dc.format.extent1609187 bytes-
dc.format.mediumapplication/pdf-
dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subject타원형 편미분방정식-
dc.subject정칙성-
dc.subject.ddc510-
dc.titleRegularity results for generalized elliptic problems in bounded domains-
dc.title.alternative유계영역에서 정의된 일반화된 타원형 방정식의 정칙성-
dc.typeThesis-
dc.description.degreeDoctor-
dc.citation.pages100-
dc.contributor.affiliation자연과학대학 수리과학부-
dc.date.awarded2017-02-
Appears in Collections:
College of Natural Sciences (자연과학대학)Dept. of Mathematical Sciences (수리과학부)Theses (Ph.D. / Sc.D._수리과학부)
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