The Boundary Harnack Principle for Nonlocal Elliptic Operators in Non-divergence Form

Potential Analysis - We prove a boundary Harnack inequality for nonlocal elliptic operators L in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that...     

线性随机微分方程的离散波形松弛方法的几乎必然收敛性

提出了随机微分方程的离散型波形松弛方法,证明了它是几乎必然收敛的.此外,通过数值实验验证了所得结果.     

Entire Solutions of Discrete Elliptic Equations

By means of the monotone methods, comparison theorems for the existence and uniqueness of solutions of discrete elliptic type partial difference equations are derived. These results enable us to find positive and bounded solutions to several specific equations.AMS Subject Classification (1991): 39A10.     

Uniform Elliptic Estimate for an Infinite Plate in Linear Elasticity

Abstract

We present a new study of linear elasticity for an infinite three-dimensional plate of finite thickness Ω = ?² × (?1, 1). We first characterize the kernel of the operator of elasticity as polynomials which can be build from the kernel of the classical Kirchhoff–Love model of plate. Using this characterization, we get optimal uniform elliptic estimates W ^{k, p} , C ^{k, α} on the solution as a function of the exterior forces. We also give an interior estimate.     

Positive Solutions for Asymptotically Linear Cone-Degenerate Elliptic Equations

In this paper, the authors study the asymptotically linear elliptic equation on manifold with conical singularities ??_Bu + λu = a(z)f(u), u ≥ 0 in R^N₊,where N = n + 1 ≥ 3, λ > 0, z = t,x₁,· · · ,x_n, and ?_B = (t?_t)^{2 + ?2_x1 + · · · + ?2_xn. Combining properties of cone-degenerate operator, the Pohozaev manifold and qualitative properties of the ground state solution for the limit equation, we obtain a positive solution under some suitable conditions on a and f.}     

Boundary Singularities in Linear Elliptic Differential Equations

Finite-difference methods are relatively inefficient in theneighbourhood of boundary singularities in elliptic problems.A combination of special treatment near the singularity, basedon local satisfaction of the differential equation and boundaryconditions, is here matched with finite-difference formulaein the rest of the field. The method is applied to a generalself-adjoint equation with either Dirichlet, Neumann or mixedconditions on parts of the boundary consisting of two straightlines meeting at the singular point. A practical problem, formerlysolved by more extensive labour, illustrates the power of themethod.     

Estimate of Condition Number for Some Discrete Ill-Posed Equations

This paper extends the numerical method and the estimate of the condition number for some discrete ill-posed equations with positive definite matrix in [2] to the case with generalized positive definite matrix.     

Asymptotics for Solutions of Elliptic Equations in Double Divergence Form

We consider weak solutions of an elliptic equation of the form ? _i ? _i (a _ij u) = 0 and their asymptotic properties at an interior point. We assume that the coefficients are bounded, measurable, complex-valued functions that stabilize as x → 0 in that the norm of the matrix (a _ij (x) ? δ _ij ) on the annulus B _2r \ B _r is bounded by a function Ω(r), where Ω²(r) satisfies the Dini condition at r = 0, as well as some technical monotonicity conditions; under these assumptions, solutions need not be continuous. Our main result is an explicit formula for the leading asymptotic term for solutions with at most a mild singularity at x = 0. As a consequence, we obtain upper and lower estimates for the L ^p -norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L ^p -mean as r → 0.     

Two-scale Convergence and Homogenization for a Class of Quasilinear Elliptic Equations

By use of Fourier analysis techniques, we obtain some new properties of the almost-periodic functions and extend the two-scale convergence method in the homogenization theory to the case of almost-periodic oscillations. Then, we use some new techniques to study the homogenization for quasilinear elliptic equations with almostperiodic coefficients: div a(x,x/ε, u, Du) = f(x) in Ω and obtain the weak convergence and corrector result.     

Convergence of Finite-Difference Schemes for Elliptic Equations with Variable Coefficients

We study the convergence of finite-difference schemes for second-orderelliptic equations with variable coefficients. We prove thatthe convergence rate in the discrete W₂¹ norm is of the orderh^{s –1} if the solution of the boundary value problem belongsto the Sobolev space W₂^s (1 < s 3).     

Irregular Solutions of Linear Degenerate Elliptic Equations

We give examples of discontinuous solutions and of unbounded solutions of linear isotropic degenerate elliptic equations. Discontinuous solutions exist even when both the maximum eigenvalue and the inverse of the minimum eigenvalue of the matrix of the coefficients are in the intersection of all the Lp spaces.     

A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numerical approximation to the gradient in a piecewise irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in the continuous Lagrange finite element space. The variational setting naturally provides an a posteriori error which can be used in an adaptive refinement algorithm. The error estimates under the $L^2$ norm and the energy norm for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.     

Elliptic Differential-Difference Equations of General Form in a Half-Space

Mathematical Notes - We study the Dirichlet problem in a half-space for elliptic differential-difference equations with operators that are compositions of differential operators and shift operators...     

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.     

Random Walks Associated with Non-Divergence Form Elliptic Equations

This paper is concerned with the study of the diffusion process associated with a nondivergence form elliptic operator in d dimensions, d2. The authors introduce a new technique for studying the diffusion, based on the observation that the probability of escape from a d–1 dimensional hyperplane can be explicitly calculated. They use the method to estimate the probability of escape from d–1 dimensional manifolds which are C ^1, , and also d–1 dimensional Lipschitz manifolds. To implement their method the authors study various random walks induced by the diffusion process, and compare them to the corresponding walks induced by Brownian motion.     

一类拟线性椭圆型方程下解的先验估计

本文在一定的结构条件下,对如下形状的拟线性椭圆型方程∫Ｇｖ·Ａ（ｘ,ｕ,ｕ）＋ｖＢ（ｘ,ｕ,ｕ）｛｝ｄｘ＝０,ｖ∈Ｗ１ｐ（Ｇ）的下解作出了新的先验估计,发展了Ｇｉｌｂａｒｇ＿Ｔｒｕｄｉｎｇｅｒ的某些结果·     

Boundary Lipschitz Regularity of Solutions for Semilinear Elliptic Equations in Divergence Form

In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C1,Dinicondition at a boundary point, and the nonhomogeneous term satisfies Dini continuity condition and Lipschitz Newtonian potential condition, then the solution is Lipschitz continuous at this point. Furthermore, we generalize this result to Reifenberg ...     

Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces

In this article, we prove optimal convergence rates results for regularization methods for solving linear ill-posed operator equations in Hilbert spaces. The results generalizes existing convergence rates results on optimality to general source conditions, such as logarithmic source conditions. Moreover, we also provide optimality results under variational source conditions and show the connection to approximative source conditions.     

Elliptic Equations with Translations of General Form in a Half-Space

Mathematical Notes - We study the Dirichlet problem in a half-space for elliptic differential-difference equations with operators representing superpositions of differential operators and...