一类再生Hilbert空间上偏微分算子的有界性

研究了一类再生Hilbert空间上偏微分算子的有界性,得到了偏微分算子有界的一个充分必要条件,推广了文献中的结果．     

一类函数空间上偏微分算子有界性和紧性的刻画(英文)

本文研究了一类具有再生核的多复变量函数空间上偏微分算子的有界性和紧性.利用算子理论的方法,获得了偏微分算子是有界的和紧的充分必要条件,推广了文献[1]中的结果.     

对《Some Notes on Solvability of LPDO》一文的补充

文[1]对具重特征的线性偏微分算子 L(x,D)=D_1~2+x_1~2D_2~2+PD_2+qD_1+c 的局部可解性作了有趣的说明。所得结果表明具重特征的线性偏微分算子中的低阶项对它的局部可解性起着关键的作用。现在我们对更一般的具重特征的偏微分算子给出类似的结论。     

海森堡群H~n上算子_μ~m的可解性和亚椭圆性

该文给出海森保群Hn上偏微分算子的显式基本解um,据此um,研究算子的可解性和亚椭圆性．     

一类非主型偏微分算子局部可解性的必要条件

<正> 一 自从H.Lewy提出了第一个不可解算子之后,偏微分算子的可解性问题受到许多人的注意.现在,对于这一课题的研究,已经取得了相当一般的结果. 一个具C~∞系数的线性偏微分算子P(x,D),我们说它在分布意义下是局部可解的,是指:在Ω中x_o∈Ω,存在x_o的一个邻域U,使得f∈C_o~∞(U),有P(x,D)u=f成立.     

期权定价的小波方法

基于Daubechies正交小波,对微分算子进行小波近似,从而求解Black-Scholes方程,为期权定价提出了一种新的尝试.通过偏微分算子和小波系数的稀疏化,相对二叉树法,大大减少了计算量,提高了运算速度.     

关于变数分离的线性偏微分方程的基本解的结构

<正> 对于相当一般的二阶线性非抛物型偏微分方程,Hadamard给出了由方程的系数构造基本解的方法.本文研究的是,当一个二阶线性偏微分算子是变数分离的两个低维的二阶线性偏微分算子之和时,它的方程的基本解与相应于分解后两个低维方程的基本解之间应有的关系.本文给出了这种关系的简明的结构式.     

二步幂零Lie群上sub-Laplace算子的谱

本文首先研究包括Ｇｒｅｉｎｅｒ算子和二步幂零Ｌｉｅ群上ｓｕｂ Ｌａｐｌａｃｅ算子Ｌ在内的广义齐次线性偏微分算子的谱，然后利用酉表示理论确定了算子Ｌ的谱结构，并深化了Ｋ．Ｆｕｒｕｔａｎｉ，Ｋ．Ｓａｇａｍｉ，Ｎ．Ｏｔｓｕｋｉ关于Ｌａｐｌａｃｅ算子的有关结果．     

一类具重特征线性偏微分算子的局部可解性

运用Ｇａｒｄｉｎｇ不等式和Ｆｅｆｅｒｍａｎ－Ｐｈｏｎｇ不等式，建立了一类具实主会征的线性偏微分算子的局部可解性。     

Heisenberg群上齐次左不变偏微分算子的相对基本解

运用映 S(R~n)到 S′(R~n)的连续线性算子的 Hermite 表示理论和 Heisenberg 群的酉表示理论,证明了当 Heisenberg 群上齐次左不变偏微分算子的群 Fourier 变换满足一定条件时,必存在相对基本解,并给出了相对基本解的计算公式.本文结果把 Greiner-Kohn-Stein 和Geller 等人关于齐次横截椭圆算子的相应结果推广到一般齐次算子.     

Existence of the Møller wave operators in quantum scattering theory

Existence of the wave operators of quantum mechanical scattering theory is established. The Hamiltonians are constant coefficient partial differential operators and perturbations of them by variable coefficient linear partial differential operators. The coefficients may be short range in the usual sense but they may also be oscillating. The problem of establishing existence is reduced to that of approximating solutions of a certain partial differential equations on cones in phase space. The proof is based on a refinement of Cook's argument. Research supported in part by a fellowship, awarded by the University of Toledo.     

New proofs and generalisations of theorems of existence and uniqueness for the goursat problem

Abstract theorems of existence and uniqueness are proved for a differential equation whose solution takes its values in a sequence of Banach spaces called a Banach filtration (a notion introduced by F. Treves). The abstract theorems are then applied to obtain existence and uniqueness theorems of a classical nature bearing on that generalization of the Cauchy problem of partial differential equations known as the Goursat problem. All the results so obtained remain true in the case when the equations involve more general operators than partial differential operators (e.g., pseudo-differential operators)     

Inverse Problems on the Least Eigenvalue

Some uniqueness theorems on the least eigenvalue are provided for wide classes of self-adjoint operators: differential operators with operator-valued potentials, higher-order partial differential operators and the p-Laplacian.     

A note on the extension of the Dirac method

In this note we extend the Dirac method to partial differential equations involving higher order roots of differential operators.     

Noether, partial Noether operators and first integrals for a linear system

We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians.     

On the approximation of stochastic partial differential equations i

The stability of abstract stochastic partial differential equations with respect to the simultaneous perturbation of the driving processes and of the differential operators is investigated. The results obtained here will be applied to concrete stochastic partial differential equations in the continuation of this paper     

Asymptotic behavior of the spectrum of differential equations

This survey is devoted to an exposition of results on the asymptotics of the discrete spectrum of self-adjoint differential operators, mainly partial differential operators.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 14, pp. 5–58, 1977.     

Bounds on the non-real spectrum of differential operators with indefinite weights

Ordinary and partial differential operators with an indefinite weight function can be viewed as bounded perturbations of non-negative operators in Krein spaces. Under the assumption that 0 and $\infty $ are not singular critical points of the unperturbed operator it is shown that a bounded additive perturbation leads to an operator whose non-real spectrum is contained in a compact set and with definite type real spectrum outside this set. The main results are quantitative estimates for this set, which are applied to Sturm–Liouville and second order elliptic partial differential operators with indefinite weights on unbounded domains.