변동지수르베그공간상의 타원형 및 포물형 방정식에 대한 대역적 그레디언트 가늠

학위논문 (박사)-- 서울대학교 대학원 : 수리과학부, 2015. 2. 변순식.이 학위 논문에서는 변동지수르베그공간에서의 발산형 타원형 및 포물형 방정식들에 대한 대역적 칼데론-지그먼드 이론에 대하여 연구한다. 특히, 적절한 가늠을 유도함으로써 디리클레 형식의 경계값이 영인 방정식의 유일한 해의 그레디언트가 변동지수르베그공간에서 비동차항과 동등한 적분가능성을 가진다는 것을 증명한다. 본 연구에서는 선형 타원형 방정식, 선형 포물형 방정식, 변동 성장조건을 가지는 타원형 방정식, 변동 성장조건을 가지는 포물형 방정식등 네가지 형태의 방정식을 다룬다. 그리고 대역적 칼데론-지그먼드 이론을 얻기위한 변동지수, 계수함수, 경계영역의 최소 조건을 제시한다.We establish global Calderón-Zygmund theory for divergence type elliptic and parabolic equations in variable exponent Lebesgue spaces. We prove that the gradient of the unique weak solution to a given problem with the zero Dirichlet boundary condition is as integrable as the nonhomogeneous term of the problem in variable exponent Lebesgue space by deriving a suitable estimate. In this thesis we consider four equations: the linear elliptic equation, the linear parabolic equation, the nonlinear elliptic equation with variable growth and the nonlinear parabolic equation with variable growth. We also provide reasonable answers to minimal regularity assumptions on the variable exponents, the coefficients and the boundary of the domain to obtain the desired Calderón-Zygmund theory.1 Introduction 1 2 Preliminaries 6 2.1 Notations 2.2 Variable exponent spaces 2.3 Technical background 3 Gradient estimates for linear equations in variable exponent spaces 3.1 W^{1,p(·)}-regularity for elliptic equations with measurable coefficients in nonsmooth domains 3.1.1 Main result 3.1.2 Proof of Theorem 3.1.4 3.2 Optimal gradient estimates for parabolic equations in variable exponent spaces 3.2.1 Main result 3.2.2 Proof of Theorem 3.2.4 4 Nonlinear elliptic equations with variable exponent growth in nonsmooth domains 4.1 W^{1,q(·)}-estimates for elliptic equations of p(x)-Laplacian type 4.1.1 Main Result 4.1.2 Auxiliary lemmas 4.1.3 Proof of Theorem 4.1.6 4.2 Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity 4.2.1 Main result 4.2.2 Proof of Theorem 4.2.1 5 Nonlinear parabolic equations with variable exponent growth in nonsmooth domains 5.1 Main Result 5.1.1 Notations and log-Holder continuity for parabolic problems 5.1.2 Main result 5.2 Preliminaries 5.2.1 Parabolic Sobolev spaces and P.D.E. with variable exponents 5.2.2 Self improving integrability 5.2.3 Lipschitz regularity 5.2.4 Technical tools 5.3 Comparison Estimates 5.4 Gradient Estimate in the Variable Exponent Lebesgue Spaces 5.4.1 choice of intrinsic cylinders 5.4.2 Comparison estimates 5.4.3 Gradient estimates on upper-level sets 5.4.4 Local gradient estimates in L^{p(·)q(·)}-space: the proof of (5.1.12) 5.4.5 Global gradient estimates in L^{p(·)q(·)}-space: the proof of (5.1.13)Docto

Finite element approximation of the p(⋅)p(\cdot)-Laplacian

We study a~priori estimates for the Dirichlet problem of the $p(\cdot)$-Laplacian, $$-\mathrm{div}(|\nabla v|^{p(\cdot)-2} \nabla v) = f.$$ We show that the gradients of the finite element approximation with zero boundary data converges with rate $O(h^\alpha)$ if the exponent $p$ is $\alpha$-H\"{o}lder continuous. The error of the gradients is measured in the so-called quasi-norm, i.e. we measure the $L^2$-error of $|\nabla v|^{\frac{p-2}{2}} \nabla v$

Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.Comment: 24 pages, 3 figure

Variable Lorentz estimate for nonlinear elliptic equations with partially regular nonlinearities

We prove global Calder\'on-Zygmund type estimate in Lorentz spaces for variable power of the gradients to weak solution of nonlinear elliptic equations in a non-smooth domain. We mainly assume that the nonlinearities are merely measurable in one of the spatial variables and have sufficiently small BMO semi-norm in the other variables, the boundary of domain belongs to Reifenberg flatness, and the variable exponents $p(x)$ satisfy log-H\"older continuity

Lorentz estimates for the gradient of weak solutions to elliptic obstacle problems with partially BMO coefficients

We prove global Lorentz estimates for variable power of the gradient of weak solution to linear elliptic obstacle problems with small partially BMO coefficients over a bounded nonsmooth domain. Here, we assume that the leading coefficients are measurable in one variable and have small BMO semi-norms in the other variables, variable exponents $p(x)$ satisfy log-H\"older continuity, and the boundary of domains are so-called Reifenberg flat. This is a natural outgrowth of the classical Calder\'{o}n-Zygmund estimates to a variable power of the gradient of weak solutions in the scale of Lorentz spaces for such variational inequalities beyond the Lipschitz domain

측도데이터를 가지는 방정식에 대한 정규화 추정값

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2018. 2. 변순식.이 학위논문에서는 매끄럽지 않은 경계를 가지는 영역 하에서 측도데이터를 가지는 비선형 타원형 및 포물형 방정식에 대한 대역적인 칼데론-지그문트 유형의 추정값에 대하여 연구한다. 비선형성과 영역의 경계에 대한 최소한의 조건하에서 해의 그래디언트가 주어진 측도데이터의 극대함수와 대역적으로 동등한 적분가능성이 가지는 것을 증명함으로써 최적의 칼데론-지그문트 유형의 추정값을 입증한다. 우리는 변수지수 성장조건을 가지는 비선형 타원형 방정식, 가측 비선형성을 가지는 타원형 및 포물형 방정식에 대하여 각각 대역적인 칼데론-지그문트 유형의 추정값을 제시한다.We establish global Calderon-Zygmund type estimates for nonlinear elliptic and parabolic equations in nonsmooth bounded domains when the right-hand side is a finite signed Radon measure. We first investigate a quasilinear elliptic equation with variable growth. We obtain an optimal global Calderon-Zygmund type estimate for such a measure data problem, by proving that the gradient of a very weak solution to the problem is as globally integrable as the first order maximal function of the associated measure, up to a correct power, under minimal regularity requirements on the nonlinearity, the variable exponent and the boundary of the domain. Secondly, we study a nonlinear elliptic equation with measurable nonlinearity. A global Calderon-Zygmund type estimate in variable exponent spaces is established under optimal regularity assumptions on the nonlinearity and the Reifenberg flatness of the boundary. We finally consider a nonlinear parabolic equation with measurable nonlinearity. Under minimal regularity requirements on the nonlinearity and the boundary of the domain, we prove a global Calderon-Zygmund type estimate in weighted Orlicz spaces. As an application we obtain such an estimate in variable exponent spaces, which gives an alternative proof for this new result in the literature.1 Introduction 1 1.1 Measure data problems 1 1.2 Calderon-Zygmund theory 5 2 Preliminaries 9 2.1 Elliptic equations 9 2.1.1 Notation 9 2.1.2 Variable exponent spaces 10 2.1.3 Reifenberg flat domains 11 2.1.4 Auxiliary results 12 2.2 Parabolic equations 14 2.2.1 Notation 14 2.2.2 Muckenhoupt weights 14 2.2.3 Weighted Orlicz spaces 15 2.2.4 Auxiliary results 16 3 Regularity estimates for elliptic measure data problems with variable growth 19 3.1 Main results 21 3.2 Comparison estimates in L1 for regular problems . 23 3.2.1 Boundary comparisons 23 3.2.2 Interior comparisons 41 3.3 Covering arguments 42 3.4 Global Calderon-Zygmund type estimates 49 4 Optimal regularity for elliptic measure data problems in variable exponent spaces 55 4.1 Main results 56 4.2 Comparison estimates for regular problems 59 4.2.1 Boundary comparisons 60 4.2.2 Interior comparisons 63 4.3 Covering arguments 65 4.4 Calderon-Zygmund type estimates 74 4.4.1 Local estimates 77 4.4.2 Global estimates 78 5 Global weighted Orlicz estimates for parabolic measure data problems: Application to estimates in variable exponent spaces 81 5.1 Main results 83 5.2 Proof of Theorem 5.1.4 85 5.2.1 Comparisons 85 5.2.2 Covering arguments 88 5.2.3 Calderon-Zygmund type estimates 91 5.3 Application 92 Bibliography 95 Abstract (in Korean) 105Docto