【作者】 韩军强；

【导师】 钮鹏程；

【作者基本信息】 西北工业大学 ， 应用数学， 2006， 博士

【摘要】 本文的研究内容主要有以下两个部分：一、Carnot群上的偏微分算子我们考虑了一类特殊的2步Carnot群—H型群上的偏微分方程。首先，我们构造出了H型群上的一类极坐标，利用它显式地计算出了H型群上球的体积和p—次Laplace算子基本解中的常数。然后，我们讨论了H型群上一些偏微分不等式弱解的不存在性。我们用“积分关系式法”证明了H型群上退化椭圆型不等式和一阶及二阶退化发展不等式弱解的不存在性。粗略地说，我们的方法基于推导出所考虑问题的可能弱解的一个合适的先验估计，而这一过程是通过仔细地选择特殊的非负试验函数和伸缩讨论来完成的。这种方法避免了使用比较原理或最大值原理，能被应用于一些更一般的问题中。接着，我们研究了Carnot群G上双非线性退化抛物型方程“一般正局部解”的不存在性。这个方程有很强的实际背景，具有重要的应用价值。特别地，我们考虑了奇异位势V=γ（|▽_GN|^p）/N^p（γ＞0）。我们的结果说明了，在可极化Carnot群上，这类方程正解的不存在性与Hardy型不等式是紧密联系在一起的。二．与广义Greiner向量场相关的偏微分算子广义Greiner向量场为（其中κ≥1）不是任何幂零Lie群的基，不具有平移不变性，也不满足Hormander条件。与之相联系的广义Greiner算子△_L不再是任何幂零Lie群上的平移不变微分算子，但它是拟齐次偏微分算子。与椭圆型方程的几何最大值原理类似的结果，在广义Greiner向量场情形下是否成立，还是一个开问题。该开问题可叙述为：设Ω（?）R²ⁿ⁺¹是一个连通有界开集，f∈L^Q（Ω），u∈L_loc^2，Q（Ω）∩C（（?）），且在Ω上满足，则存在一个正常数C=C（Q，v，Ω），使得（?） u≤（?） u⁺+C‖f‖_L^Q（Ω）。我们通过构造与广义Greiner向量场相联系的一类非散度型方程的非平凡解，指出在广义Greiner向量场情形下，在上式的右端中，f的L^Q模是最佳可能的指标。我们建立了广义Greiner向量场的Hardy型不等式和Rellich型不等式。Hardy型不等式和Rellich型不等式在偏微分方程的研究中都有着重要作用。我们通过构造合适的辅助函数，给出了空间R²ⁿ⁺¹中拟球域内外的Hardy型不等式；给出了相应于广义Greiner向量场的p一次Laplace算子△_L，p的广义Picone型恒等式，并利用它得到了更一般的Hardy型不等式；证明了广义Greiner算子△_L的Hardy型不等式中的常数是最佳的；给出了一个具有紧支集的光滑函数的表示公式，然后建立了广义Greiner算子的一类含部分变量的Hardy型不等式，并证明了其中常数的最佳性；研究了与广义Greiner向量场有关的一些改进Hardy型不等式和Rellich型不等式；得到了一些带有余项的Hardy型不等式和Rellich型不等式。更多还原

【Abstract】 The thesis consists of two parts.一. PDOs on Carnot groupsPDEs on H-type groups which are a special class of Carnot groups of step two are considered. First, we construct the polar coordinates for H-type groups. Using this we explicitly compute the volume of balls in the sense of the distance and the constant in the fundamental solution of the p-sub-Laplacian on the N-type group. Then we discuss nonexistence of weak solutions for some partial differential inequalities on H-type groups. We prove nonexistence results of weak solutions for a degenerate elliptic inequality, a degenerate first order evolution inequality and a degenerate second order evolution inequality on the H-type group by the method of integral relations. Roughly speaking, this approach is based on the derivation of suitable a priori bounds on the possible weak solutions of the problem under consideration by carefully choosing special nonnegative test functions and scaling argument. It avoids use of comparison or maximum principle arguments and is useful in the study of some general problems.In succession nonexistence of "general positive local solutions" for a doubly nonlinear degenerate parabolic equation on Carnot group G is studied. This equation has very strong real background and important application value. In particular we take into account the singular potential V=γ|▽_GN|^p／N^p（γ＞0）. Our results indicate that nonexistence of positive solutions to this equation is intimately related to Hardy’s inequality on polarizable Carnot groups.二. PDOs related to generalized Greiner vector fieldGeneralized Greiner vector field is k≥1, which is not the basis for any nilpotent Lie gronp, does not possesses the translation invariance and does not satisfies Hormander’s hypoellipticity condition. Generalized Greiner operatorΔ_L related to it is no longer the translation invariante differential operator on any nilpotent Lie group, but is a quasihomogeneous PDO.It is still an open problem whether the result similar to the geometric maximum principle for elliptic equations holds in the situation of generalized Greiner vector field. This open problem can be stated as follows: LetΩ（?）R²ⁿ⁺¹ be a connected, bounded open set and f∈L^Q（Ω）. Suppose that u∈L_loc^{2, Q} （Ω）∩C（（?）） satisfies Lu=sum from i,j=1 to 2n a_iju,_ij≥f inΩ. There exists a constant C=C （Q, v,Ω）＞0 such that （?） u≤（?） u⁺ + C‖f‖_L^Q（Ω）. Through constructing solutions to nondivergence form equations associated to generalized Greiner vector field, we point out that in this case the L^Q norm of f in the right-hand side is the best possible character.Hardy type inequalities and Rellich type inequalities for generalized Greiner vector field are established. They play important roles in the study of PDEs. We yield some Hardy type inequalities on the domain in R²ⁿ⁺¹ by constructing suitable auxiliary functions; we give generalized Picone type identities for p—subLaplacianΔ_{L, p} related to generalized Greiner vector field and applying it get a generalized Hardy type inequality; we prove that the constant is sharp in the Hardy type inequality of generalized Greiner operatorΔ_L. A representation formula for smooth functions with compact support is found. Then a Hardy type inequality involving part variables is established and the optimality of the constant is proved. Finally we devote to researching improved Hardy type inequalities and Rellich type inequalities associated to generalized Greiner vector field. Some Hardy type inequalities and Rellich type inequalities with remainder terms are gained by improving inequalities established.更多还原