各项异性椭圆方程基本解的存在性

证明了右端可测的各项异性椭圆方程基本解的存在性,其中应用了各项异性Sobolev空间和Lebesgue空间.首先得到近似方程的解,然后通过对这些解的子列取极限,得到原方程的解.关键是要有一个近似函数空间以及近似方程的先验估计.最后运用Vitali定理证明了原方程基本解的存在性,推广和改进了已有方程.     

超双曲型方程的解的性质与 Dirichlet 问题

本文主要通过基本解、基本公式讨论了超双曲型方程的解的性质;提出超双曲型方程具非解析的广义势解;并得到超双曲型方程 Dirichlet 问题的解的表达式.特别指出:通过基本解研究超双曲型方程是自然的途径.     

六顶角带色参数杨-Baxter方程的解

通过5个解变换,3组非退化基本解和若干组退化基本解,给出了精确可解统计模型理论中六顶角带色参数杨-Baxter方程的全部解,并说明由六顶角带色参数杨-Baxter方程的解可以得到六顶角带谱参数杨-Baxter方程的全部解.     

高阶阻尼波动方程的一些估计

研究了高阶阻尼波动方程在L~p中的一些估计,利用调和分析的方法与工具,尤其是振荡积分的方法,得到了方程基本解对应的核的点态估计以及基本解的时空估计.并利用这些估计,给出了方程全局解的一个结果.     

Keldysh型算子的基本解

本文研究Keldysh型算子Lαu=△δ^2u/δx^2＋yδ^2u/δy^2＋αδu＊δy-与Tricomi算子不同的另一类基本的混合型算子-的基本解．得到了α〉1/2时Keldysh型算子基本解的显式表示．这类基本解一般比Tricomi算子的基本解具有更强的奇性．当α〈1/2时Keldysh型算子的基本解需用发散积分的有限部分来表示．     

板弯曲问题的具两组高阶基本解序列的MRM方法

讨论了双参数地基上薄板弯曲问题.利用两组高阶基本解序列,即调和及重调和基本解序列,采用多重替换方法(MRM方法),得到了板弯曲问题的MRM边界积分方程.证明了该方程与边值问题的常规边界积分方程是一致的.因此由常规边界积分方程的误差估计即可得到板弯曲问题MRM方法的收敛性分析.此外该方法还可推广到具多组高阶基本解序列的情形.     

Tricomi算子的基本解

考虑含三个自变量的Tricomi方程Tu=y(u_(x_1x_1)+u_(x_2x_2))+u_(yy)=0 (1)奇点为(a,b,0)的基本解.相对于两维的Tricomi方程,由于其奇性的增强,用通常的分布论计算基本解时,得到的积分发散,以致无法用该方法得到基本解,此时有必要引入散度积分主部来定义分布论中的基本解.我们利用特征线法在Cauchy主值意义下求得其基本解.     

一类反应扩散方程组的初值问题解的渐近性质

本文利用基本解的估计式及其积分的某些渐近性质,得到了一类反应扩散方程组的初值问题在 t→∞时解的渐近性质。     

横观各向同性压电材料的基本解

推导得到一组在体积力作用下压电材料平衡方程的一般解,对横观各向同性压电材料,利用一般解结合体积势理论及构造一类调和函数的方法,得到了无限体在集中力和点电荷作用下的位移和电势的有限形式的表达式,从而给出了边界元法中可用的基本解.     

n阶脉冲微分方程边值问题

本文利用微分不等式原理及脉冲微分方程初值问题基本理论研究了ｎ类ｎ阶脉冲微 分方程边值问题,得到了该边值问题解的存在性及解的存在唯一性的新的结果．     

Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ${\Bbb G}We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ? be any group of Heisenberg type whose Lie algebra is g enerated by m left invariant vector fields and with a q-dimensional center. Let and Then, with A _Q as the sharp constant, where ∇_? denotes the subellitpic gradient on ? This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18]. Received March 15, 2001, Accepted September 21, 2001     

On fundamental solution for powers of the sub-Laplacian on the Heisenberg group

We discuss the fundamental solution for m-th powers of the sub-Laplacian on the Heisenberg group. We use the representation theory of the Heisenberg group to analyze the associated m-th powers of the sub-Laplacian and to construct its fundamental solution. Besides, the series representation of the fundamental solution for square of the sub-Laplacian on the Heisenberg group is given and we also get the closed form of the fundamental solution for square of the sub-Laplacian on the Heisenberg group with dimension n=2,3,4.     

四元Heisenberg群上次拉普拉斯算子的m幂次的基本解

本文研究了四元Heisenberg群上次拉普拉斯算子的m幂次的基本解，该结论是Heisenberg群上结果的推广.本文利用了四元Heisenberg群上的Fourier变换理论构造了该群上次拉普拉斯算子的m幂次的基本解，并且给出了基本解的积分表示.     

四元Heisenberg群上次拉普拉斯算子的m幂次的基本解

本文研究了四元Heisenberg群上次拉普拉斯算子的m幂次的基本解,该结论是Heisenberg群上结果的推广.本文利用了四元Heisenberg群上的Fourier变换理论构造了该群上次拉普拉斯算子的m幂次的基本解,并且给出了基本解的积分表示.     

Pizzetti’s Formula for H-type Groups

Let ℍ be a H-type group. We provide a generalization of Pizzetti’s Formula for an orthogonal sub-Laplacian Δ_ℍ on ℍ. A formula expressing the k-th power of the operator Δ_ℍ is also proved. These results improve those contained in a former paper by the author, Bonfiglioli (Potential Anal 17:165–180, 2002).     

A Note on Hermite and Subelliptic Operators

In this note, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point y∈R^n. We also apply this result to obtain the fundamental solutions for the Grushin operator in R^2 and the sub-Laplacian in the Heisenberg group Hn.     

The heat kernel on H-type groups

In this paper we present an explicit calculation of the heat kernel for the sub-Laplacian on an H-type group by using irreducible unitary representations of and the heat kernel for the associated Hermite operator.

     

Weighted Hardy and Rellich inequality on Carnot groups

We prove some weighted Hardy and Rellich inequalities on general Carnot groups with weights associated to the norm constructed by Folland’s fundamental solution of the Kohn sub-Laplacian.     

Spectral transform for the sub-Laplacian on the Heisenberg group

We continue our analysis of nilpotent groups related to quantum mechanical systems whose Hamiltonians have polynomial interactions. For the spinless particle in a constant external magnetic field, the associated nilpotent group is the Heisenberg group. We solve the heat equation for the Heisenberg group by diagonalizing the sub-Laplacian. The unitary map to the Hilbert space in which the sub-Laplacian is a multiplication operator with positive spectrum is given. The spectral multiplicity is shown to be related to the irreducible representations of SU(2). A Lax pair, generated from the Heisenberg sub-Laplacian, is used to find operators unitarily equivalent to the sub-Laplacian, but not arising from the SL(2,R) automorphisms of the Heisenberg group. Department of Mathematics, supported in part by NSF. Department of Physics and Astronomy, supported in part by DOE.     

Carnot Geometry and the Resolvent of the Sub-Laplacian for the Heisenberg Group

Abstract

We obtain an explicit representation formula for the sub-Laplacian on the isotropic, three-dimensional Heisenberg group. Using the formula we obtain themeromorphic continuation of the resolvent to the logarithmic plane, the existence of boundary values in the continuous spectrum, and semiclassical asymptotics of the resolvent kernel. The asymptotic formulas show the contribution of each Hamiltonian path in Carnot geometry to the spatial and high-energy asymptotics of the resolvent (convolution) kernel for the sub-Laplacian.