The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

Convergence of Finite Volume Schemes for Hamilton-Jacobi Equations with Dirichlet Boundary Conditions

We study numerical methods for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. We first propose a new class of abstract monotone approximation schemes and get a convergence rate of 1/2 . Then, according to the abstract convergence results, by newly constructing monotone finite volume approximations on interior and boundary points, we obtain convergent finite volume schemes for time-dependent Hamilton-Jacobi equations with weak Dirichlet boundary conditions. Finally give some numerical results.     

Heat Content Asymptotics for Riemannian Manifolds with Zaremba Boundary Conditions

The existence of a full asymptotic expansion for the heat content asymptotics of an operator of Laplace type with classical Zaremba boundary conditions on a smooth manifold is established. The first three coefficients in this asymptotic expansion are determined in terms of geometric invariants; partial information is obtained about the fourth coefficient.      

非齐次边界条件泛定方程的代换选择

对具有非齐次边界条件的泛定方程齐次化过程中代换的选择进行研究和探讨.基于一些相关结论和齐次化的定义得出新的研究成果,即给出对三类非齐次边界条件齐次化都适用的代换W(x,t)=A(t)x<'3>+B(t).     

On Rate-Type Viscoplastic Problems with Linear Boundary Conditions

In this paper we introduce the abstract concept of “linear boundary conditions” in the study of deformable bodies. We establish two existence and uniqueness results concerning respectively quasistatic and dynamic problems involving such type of boundary conditions. We also apply these existence results in the study of viscoplastic problems involving classical boundary conditions.     

On Nonhomogeneous Slip Boundary Conditions for 2D Incompressible Exterior Fluid Flows

The paper examines the issue of existence of solutions to the steady Navier-Stokes equations in an exterior domain in ℝ². The system is studied with nonhomogeneous slip boundary conditions. The main results proves the existence of weak solutions for arbitrary data.      

Heat Content Asymptotics with Inhomogeneous Neumann and Dirichlet Boundary Conditions

Let M be a compact manifold with smooth boundary. We establish the existence of an asymptotic expansion for the heat content asymptotics of M with inhomogeneous Neumann and Dirichlet boundary conditions. We prove all the coefficients are locally determined and determine the first several terms in the asymptotic expansion.     

Lower Bound Estimate of Blow Up Time for the Porous Medium Equations under Dirichlet and Neumann Boundary Conditions

In this paper, we establish the lower bounds estimate of the blow up time for solutions to the nonlocal cross-coupled porous medium equations with nonlocal source terms under Dirichlet and Neumann boundary conditions. The results are obtained by using some differential inequality technique.     

On graphs with three distinct Laplacian eigenvalues

In this paper,an equivalent condition of a graph G with t(2≤t≤n)distinct Laplacian eigenvalues is established.By applying this condition to t=3,if G is regular(neces- sarily be strongly regular),an equivalent condition of G being Laplacian integral is given.Also for the case of t=3,if G is non-regular,it is found that G has diameter 2 and girth at most 5 if G is not a tree.Graph G is characterized in the case of its being triangle-free,bipartite and pentagon-free.In both cases,G is Laplacian integral.     

具有非局部边界条件的奇摄动反应扩散问题

本文研究了一类具有非局部边界条件的奇摄动反应扩散初始边值问题．在适当的条件下，利用比较定理讨论了问题解的渐近性态．     

Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.     

常型Sturm-Liouville问题的左定边值条件

本文刻画了常型Sturm-Liouville问题的左定边值条件．通过Sturm-Liouville微分算式的系数、区间端点以及边值条件给出了其左定性的充要条件．应用自伴边值条件分类,确切地给出了所有可能的左定边值条件．     

四阶正则微分算子耦合自共轭边界条件的基本标准型

本文利用新的方法给出了4阶正则微分算子耦合自共轭边界条件的基本标准型,新标准型中的4个分块小矩阵为对称矩阵,且其行列式的模为1.这与2阶微分算子耦合边界条件的标准型极为类似,这为给出一般的高阶微分算子自共轭边界条件标准型提供了新的思路.     

全无赘数为零的正则图的结构

全无赘数irt是图的一个重要参数.本文对irt=0的正则图的结构进行了探讨,提供了构造irt=0的正则图的一个方法.     

Robin Approximation of Dirichlet Boundary Value Problems

This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L²(?Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H¹ and L² norms are shown to have convergence rates that depend on the regularity of the Dirichlet data.     

On the Existence and the Uniform Decay of a Hyperbolic Equation with Non-Linear Boundary Conditions

In this paper, we study a hyperbolic model based on the equation with nonlinear boundary conditions given by .We prove the existence and the uniqueness of global solutions. Also, we obtain the uniform decay of the energy without control of its derivative sign.AMS Subject Classification (2000), 35L05, 35L70, 35B40     

可化约拟线性双曲组带一类非局部边界条件的精确边界能控性

先证明了一类非线性积分方程解的存在唯一性,利用此结果,建立了可化约拟线性双曲组带一类非局部边界条件的单侧精确边界能控性.     

Three-Dimensional Boundary Value Problems of Elastothermodiffusion with Mixed Boundary Conditions

We investigate the basic boundary value problems of the connected theory of elastothermodiffusion for three-dimensional domains bounded by several closed surfaces when the same boundary conditions are fulfilled on every separate boundary surface, but these conditions differ on different groups of surfaces. Using the results of papers [1–8], we prove theorems on the existence and uniqueness of the classical solutions of these problems.