一类变系数亚椭圆算子的亚椭圆性边值问题

本文研究了一类在边界附近为定强算子的变系数亚椭圆算子的亚椭圆性边值问题。首先讨论了一个半空间R~+_n中的变系数亚椭圆算子,当其在B~0_n附近是定强算子时,为保证半空间中的边值问题是亚椭圆性边值问题时边界算子的给法的一个充分条件,并证明在此条件下,当主算子有一个低阶项的摄动时仍为一亚椭圆性边值问题。进而,证明了R~+_n中的变系数亚椭圆算子,若它在R~0_n附近是定强的且关于D_n的系数是非零无穷次光滑函数,则其边值问题是亚椭圆性边值问题.     

一类渐近线性椭圆问题解的存在性

本文研究了一类渐近线性椭圆问题解的存在性.通过环绕定理,得到了此问题存在一个非平凡解.

Abstract：

In this article,we study the existence of solutions to an asymptotically linear elliptic problem.By the linking theorem,we obtain that such problem has a nontrivial solution.     

一类带Neumann边界条件的半线性椭圆方程正解的存在性及解的性质

研究了一类带Neumann边界条件的半线性椭圆方程.运用无(ps)条件的Mountain Pass引理证明了该方程正解的存在性,然后借助RN空间上解的性质得到了该方程解的最大值点仅有一个且在边界上取到这一性质,推广了一些已知结果.     

多椭圆孔有限大复合材料层板的热弹性分析

本文利用各向异性体平面热传导，热弹性理论中的复势方法，以保角映射，Ｆａｂｅｒ级数展开以及最小二乘边界配置技术为工具，导出了内边界条件精确满足，外边界条件近似满足的多椭圆孔复合材料层板的热传导以及热弹性问题的级数解，详细探讨了层板大小，孔径，相对孔距，孔的设置方式，椭圆度以及层板的铺层比例诸参数的影响规律，得到了一些有益结论。     

含距离位势的拟线性椭圆方程解的存在性

变分原理证明了一类含距离位势的拟线性椭圆方程齐次Dirichlet边界条件下第一特征值问题的可解性.进一步,利用临界点理论得到了一类含距离位势的非线性椭圆方程非平凡解的存在性.     

Robin特征值问题与含边界项的Hardy型积分不等式

在H~1(Ω)中,基于紧性原理和变分方法,讨论Robin边界条件下椭圆特征值问题的解,获得了一个新的带边界项的Hardy型不等式.     

关于特征情形的部分亚椭圆定理及其应用

<正> 边值问题的适定性和其解的正则性有紧密的联系.在研究边值问题解到边界的正则性时,Hrmander([2])的部分亚椭圆定理起着十分重要的作用.在研究蜕缩椭圆、双曲和混合型算子的边值问题时,提出了特征情形下的部分亚椭圆定理的问题.考虑Ω=R~(n-1)X(0,1)上的一阶算子方程.     

椭圆外区域上Helmholtz问题的耦合法

以椭圆外区域上Helmholtz方程为例,研究一种带有椭圆人工边界的自然边界元与有限元耦合法,给出了耦合变分问题的适定性及误差分析并给出数值例子.理论分析及数值结果表明,用方法求解椭圆外问题是十分有效的.为求解具有长条型内边界外Helmholtz问题提供了一种很好的数值方法.     

Heisenberg群上次椭圆p-Laplace方程的边界估计

研究次椭圆p-Laplace方程(P＞1)解的边界性质,通过建立Heisenberg群上带有区域内点到边界Carnot-Carath閛dory距离函数的Hardy型不等式,给出了有界域上次椭圆p-Laplace方程以及带非平凡位势的次椭圆p-Laplace方程的解在边界附近的若干估计.     

含梯度项的椭圆方程组的边界爆破解

研究了含梯度项的椭圆方程组的边界爆破解的性质,其中权函数a(x),b(x)为正并且满足一定的条件.利用上下解的方法及比较原则证明了正解的存在性与唯一性,并得到了边界爆破速率的估计.     

Overdetermined Elliptic Systems

We consider linear overdetermined systems of partial differential equations. We show that the introduction of weights classically used for the definition of ellipticity is not necessary, as any system that is elliptic with respect to some weights becomes elliptic without weights during its completion to involution. Furthermore, it turns out that there are systems which are not elliptic for any choice of weights but whose involutive form is nevertheless elliptic. We also show that reducing the given system to lower order or to an equivalent system with only one unknown function preserves ellipticity.     

Higher-order finite volume methods for elliptic boundary value problems

This paper studies higher-order finite volume methods for solving elliptic boundary value problems. We develop a general framework for construction and analysis of higher-order finite volume methods. Specifically, we establish the boundedness and uniform ellipticity of the bilinear forms for the methods, and show that they lead to an optimal error estimate of the methods. We prove that the uniform local-ellipticity of the family of the bilinear forms ensures its uniform ellipticity. We then establish necessary and sufficient conditions for the uniform local-ellipticity in terms of geometric requirements on the meshes of the domain of the differential equation, and provide a general way to investigate the mesh geometric requirements for arbitrary higher-order schemes. Several useful examples of higher-order finite volume methods are presented to illustrate the mesh geometric requirements.     

Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy s = (s_y,s_ù){\sigma=(\sigma_\psi,\sigma_\wedge)} , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol s_ù{\sigma_\wedge} which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems.     

Edge quantisation of elliptic operators

The ellipticity of operators on a manifold with edge is defined as the bijectivity of the components of a principal symbolic hierarchy , where the second component takes values in operators on the infinite model cone of the local wedges. In the general understanding of edge problems there are two basic aspects: Quantisation of edge-degenerate operators in weighted Sobolev spaces, and verifying the ellipticity of the principal edge symbol which includes the (in general not explicitly known) number of additional conditions of trace and potential type on the edge. We focus here on these questions and give explicit answers for a wide class of elliptic operators that are connected with the ellipticity of edge boundary value problems and reductions to the boundary. In particular, we study the edge quantisation and ellipticity for Dirichlet–Neumann operators with respect to interfaces of some codimension on a boundary. We show analogues of the Agranovich–Dynin formula for edge boundary value problems. Nicoleta Dines and Bert-Wolfgang Schulze were supported by Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany. Xiaochun Liu was supported by NNSF of China through Grant No. 10501034, and Chinese-German Cooperation Program “Partial Differential Equations”, NNSF of China and DFG of Germany.     

Noncoercive Boundary Value Problems for Elliptic Partial Differential and Differential-Operator Equations

In this paper we find conditions guarantee that irregular boundary value problems for elliptic differential-operator equations of the second order in an interval are fredholm. We apply this result to find some algebraic conditions guarantee that irregular boundary value problems for elliptic partial differential equations of the second order in cylindrical domains are fredholm. Apparently this is the first paper where the regularity of an elliptic boundary value problem is not satisfied on a manifold of the dimension equal to dimension of the boundary. Nevertheless the problem is fredholm and the resolvent is compact. It is interesting to note that the considered boundary value problems for elliptic equations in a cylinder being with separating variables are noncoercive.     

Elliptic systems of partial differential equations and the finite element method

An existence and uniqueness theorem is established for finite element solutions of elliptic systems of partial differential equations. To establish this result, an extension of Gårding's inequality is obtained which is valid for functions that do not necessarily vanish on the boundary of the region. To accomplish this extension, a stronger ellipticity condition, called very strong ellipticity, is defined with various necessary and sufficient conditions given.     

The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Summary Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.     

On coercivity and regularity for linear elliptic systems

We introduce a nonlinear method to study a ??universal?? strong coercivity problem for monotone linear elliptic systems by compositions of finitely many constant coefficient tensors satisfying the Legendre?CHadamard strong ellipticity condition. We give conditions and counterexamples for universal coercivity. In the case of non-coercive systems we give examples to show that the corresponding variational integral may have infinitely many nowhere C ¹ minimizers on their supports. For some universally coercive systems we also present examples with affine boundary values which have nowhere C ¹ solutions.     

Existence and uniqueness of large solutions for a class of non-uniformly elliptic semilinear equations

In this paper, we study existence, uniqueness, and asymptotic behavior of large solutions of second-order degenerate elliptic semilinear problems in non-divergence form. The main particularity of the problem is the interior uniform ellipticity of the equation, which degenerates on the boundary, involving an effect on the boundary blow-up profile of the solution.