Linear elliptic boundary value problems in varying domains

The paper is devoted to the study of solutions to linear elliptic boundary value problems in domains depending smoothly on a small perturbation parameter. To this end we transform the boundary value problem onto a fixed reference domain and obtain a problem in a fixed domain but with differential operators depending on the perturbation parameter. Using the Fredholm property of the underlying operator we show the differentiability of the transformed solution under the assumption that the dimension of the kernel does not depend on the perturbation parameter. Furthermore, we obtain an explicit representation for the corresponding derivative.     

Boundary value problems with causal operators

In this paper, we apply the monotone iterative method for nonlinear two-point boundary value problems with causal operators. We formulate sufficient conditions under which such problems have extremal or quasisolutions in a corresponding sector. We also investigate differential inequalities.     

Dirichlet boundary conditions for elliptic operators with unbounded drift

We study the realisation of the operator in with Dirichlet boundary condition, where is a possibly unbounded open set in , is a semi-convex function and the measure lets be formally self-adjoint. The main result is that at is a dissipative self-adjoint operator in .

     

Nonlinear eigenvalue problems for quasilinear operators on unbounded domains

We prove several existence results for eigenvalue problems involving the p-Laplacian and a nonlinear boundary condition on unbounded domains. We treat the non-degenerate subcritical case and the solutions are found in an appropriate weighted Sobolev space. Received May 2000     

具有共振的边值问题解的存在性

利用Mawhin的重合度理论,研究具有共振的n-阶m-点边值问题x~((n))(t)=f(t,x(t),x′(t),…,x~((n-1))(t)),t∈(0,1)x(0)=x(η),x′(0)=x″(0)=…=x~((n-2))(0)=0,x~((n-1))(1)=α_ix~((n-1))(ξ_i)解的存在性,其中n≥2,m≥3,f:[0,1]×R~n→R将有界集映为有界集,且当x(t)∈C~(n-1)[0,1]时,f(t,x(t),x′(t),…,x~((n-1))(t))∈L~1[0,1],0<ξ_1<ξ_2<…<ξ_(m-2)<1,0<η<1,α_i∈R.在这里并不要求f具有连续性.     

m点边值共振问题解的存在性与上下解方法

研究一类共振情形下二阶m点边值问题(ρ(t)x′)′=f(t,x(t),x′(t)),t∈[0,1],x′(0)=0,x(1)=∑m-2i=1αix(ηi),其中mi 3为整数,αi 0,ηi∈(0,1)(i=1,2,…,m-2)为常数,满足∑m-2i=1αi=1,0<η1<η2<…<ηm-2<1.本文的研究工具主要依赖于一个新的增算子不动点定理,本质不同于以往文献中使用的Mawhin重合度定理.     

On the structure of the solution of singularly perturbed initial boundary value problems with an unbounded spectrum of the limit operator

Singularly perturbed initial boundary value problems are studied for some classes of linear systems of ordinary differential equations on the semiaxis with an unbounded spectrum of the limit operator. We give a new version of the proof of the existence of a unique and bounded (as ε→+0) solution for which with the help of the splitting method we construct a uniform asymptotic expansion on the entire semiaxis and describe all singularities (reflecting the structure of the corresponding boundary layers) in closed analytic form, including the critical case in which the points of the spectrum of the limit operator can touch the imaginary axis; this supplements previous results. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 831–835, June, 1999.     

共振条件下积分边值问题解的存在性

运用Mawhin重合度理论建立了二阶Stieltjes积分边值问题解的存在性定理,其所得结果推广了多点边值问题已有的一些结论。     

Integral estimates of the solutions to the helmholtz equation in unbounded domains

The following boundary value problem is studied:

一类具二维核共振条件三阶多点边值问题解的存在性

利用Mawin的重合度定理建立了一类三阶多点边值问题边值条件解的存在性.同时给出一个例子以说明该存在性结果的可用性.     

hypergenic函数的边值问题

主要研究Clifford分析中hypergenic函数的边值问题A(y)Ψ*+f(y)+B(y)Ψ*-f(y)=G(y)L(Ψ*+f(y),Ψ*-f(y)).首先讨论hypergenic拟Cauchy型积分的相关性质;其次利用Schauder不动点原理证明了非线性边值问题解的存在性;最后利用压缩映射原理证明了线性边值问题解的存在唯一性.     

Noncontinuous data boundary value problems for Schrödinger equation in Lipschitz domains

The noncontinuous data boundary value problems for Schrödinger equations in Lipschitz domains and its progress are pointed out in this paper. Particularly, the L ^p boundary value problems with p > 1, and H ^p boundary value problems with p < 1 have been studied. Some open problems about the Besov-Sobolev and Orlicz boundary value problems are given.     

Boundary problems with meromorphic symbols in cylindrical domains

We show relative index formulas for boundary value problems in cylindrical domains and Sobolev spaces with different weights at ±∞. The amplitude functions are meromorphic in the axial covariable and take values in the space of boundary value problems on the cross section of the cylinder. Copyright © 2005 John Wiley & Sons, Ltd.     

Mixed boundary value problems for nonlinear elliptic equations in multidimensional non-smooth domains

The nonlinear elliptic equation is investigated. It is supposed that u fulfils a mixed boundary value condition and that Ω ? IRⁿ (n ≥ 3) has a piecewise smooth boundary. W^s,2 — regularity (s < 3/2) of u and L^p — properties of the first and the second derivatives of u are proven.     

具有共振的2n阶m点边值问题的可解性

对具有共振的高阶多点边值问题进行研究.首先在具有2n-1阶连续导数的函数全体所成的空间X的子集上定义了指数为0的Fredholm算子L,并在X上定义了投影算子P,使得算子L在其定义域和P的核的交集上是可逆的.然后,在Lebesgue可积函数全体所成的空间Y上定义了投影算子Q,使得L的逆与I-Q及非线性项f的复合是紧算子,其中,I是Y上的恒同算子.最后通过赋予f一定的增长条件,利用Mawhin的重合度理论,证明了具有共振的2n阶m点边值问题至少存在一个解,并给出一个例子验证这一结果.在这里不要求f具有连续性.