Carnot群上严格非正态极值的存在性

设G是一个Carnot群,D={e1,e2}是G上一个左不变括号生成分布.在本文中,构造存在一类维数大于5的Carnot群,其上存在严格非正态极值.从而可知Gole,Karidi构造的第一个存在奇异测地线的Carnot群只是本文的一个特殊例子.     

拟线性次椭圆方程很弱解的正则性

该文利用扰动向量场的Hodge分解及估计构造关于弱解X梯度场的反向Hlder不等式,从而建立了Carnot群上的散度型拟线性次椭圆方程很弱解梯度的可积性指数的提高,得到其很弱解是经典意义的弱解结论.     

Heisenberg群上移动球面法的应用——一类半线性方程的Liouville型定理

作为Heisenberg群上移动球面法的基础,在Heisenberg群上引入了一类CR反演变换.作为应用,讨论了Heisenberg群上的次临界方程,证明任意非负柱对称解均为平凡解.     

Carnot群上的Hardy型不等式和唯一延拓性

给出了一般Carnot群上的L~p加权Hardy型不等式,1p+∞,由此导出了Hardy-Sobolev型不等式.作为对该不等式的一个应用,考虑了一类带奇异位势的p-sub-Laplace方程,对其弱解建立了二重性和唯一延拓性.     

退化Sobolev-Galpern 方程有界解的保持性

在这篇文章中, 我们考虑了一类退化Sobolev-Galpern 方程有界解的保持性. 我们先研究了线性Sobolev-Galpern 方程的解的存在性、唯一性, 并由其解定义出了一簇发展算子. 在此基础上, 我们研究了发展算子的指数二分性、非齐次线性方程有界解的Fredholm 更替. 最后, 我们将此结果用以研究退化Sobolev-Galpern 方程在非自治扰动下有界解的保持性, 并给出了保持的条件.     

退化Sobolev-Galpern方程有界解的保持性 谨以此文致《中国科学》创刊六十周年

在这篇文章中,我们考虑了一类退化Sobolev-Galpern方程有界解的保持性.我们先研究了线性Sobolev-Galpern方程的解的存在性、唯一性,并由其解定义出了一簇发展算子.在此基础上,我们研究了发展算子的指数二分性、非齐次线性方程有界解的Fredholm更替.最后,我们将此结果用以研究退化Sobolev-Galpern方程在非自治扰动下有界解的保持性,并给出了保持的条件.     

RN中含Φ-Laplace算子和凹凸非线性项的拟线性椭圆型方程解的存在性

该文研究了全空间中一类含Φ-Laplace算子和凹凸非线性项的拟线性椭圆型方程非平凡解的存在性和多重性.利用Nehari流形方法和纤维映射等技巧,在参数较小的情况下,得到方程至少有两个非平凡解,其中一个是基态解.     

关于齐次Carnot群上广义Morrey空间中一些性质（英文）

本文研究了关于Heisenberg群上的广义Morrey空间和Carnot群上的Lebesgue空间中Riesz位势算子或者分数阶极大算子的行为.根据Heisenberg群中抽象调和分析方法以及sub Laplacian算子的Dirichlet问题解的表示公式,本文主要给出了关于齐次Carnot群G上消失的广义Morrey空间V L~(p,?)(G)中的加权Hardy算子、分数阶极大算子和分数阶位势算子的有界性刻画.进而也得到无消失模的广义Morrey空间上Morrey位势的浸入不等式.所有这些结果推广了关于Heisenberg群上的广义Morrey空间和Carnot群上的Lebesgue空间中的相关结论.     

混合局部-非局部椭圆方程奇异解的单调性和对称性

该文运用移动平面法研究了一类混合局部-非局部半线性椭圆方程奇异解的单调性和对称性.     

外部区域上的一类半线性伪抛物型方程

本文主要考虑在外区域上的一类半线性伪抛物型方程的解的长时间行为.本文得到的临界指标揭示了边界上的非齐次条件将对解的长时间行为产生本质影响.     

A sufficient condition for nonrigidity of Carnot groups

In this article we consider contact mappings on Carnot groups. Namely, we are interested in those mappings whose differential preserves the horizontal space, defined by the first stratum of the natural stratification of the Lie algebra of a Carnot group. We give a sufficient condition for a Carnot group G to admit an infinite dimensional space of contact mappings, that is, for G to be nonrigid. A generalization of Kirillov’s Lemma is also given. Moreover, we construct a new example of nonrigid Carnot group. This research was partly supported by the Swiss National Science Foundation. The author would like to thank H. M. Reimann for the helpful advices and the constant support.     

Large time behavior for the heat equation on Carnot groups

We first generalize a decomposition of functions on Carnot groups as linear combinations of the Dirac delta and some of its derivatives, where the weights are the moments of the function. We then use the decomposition to describe the large time behavior of solutions of the hypoelliptic heat equation on Carnot groups. The solution is decomposed as a weighted sum of the hypoelliptic fundamental kernel and its derivatives the coefficients being the moments of the initial datum.     

Tangent maps and tangent groupoid for Carnot manifolds

This paper studies the infinitesimal structure of Carnot manifolds. By a Carnot manifold we mean a manifold together with a subbundle filtration of its tangent bundle which is compatible with the Lie bracket of vector fields. We introduce a notion of differential, called Carnot differential, for Carnot manifolds maps (i.e., maps that are compatible with the Carnot manifold structure). This differential is obtained as a group map between the corresponding tangent groups. We prove that, at every point, a Carnot manifold map is osculated in a very precise way by its Carnot differential at the point. We also show that, in the case of maps between nilpotent graded groups, the Carnot differential is given by the Pansu derivative. Therefore, the Carnot differential is the natural generalization of the Pansu derivative to maps between general Carnot manifolds. Another main result is a construction of an analogue for Carnot manifolds of Connes' tangent groupoid. Given any Carnot manifold $(M,H)$ we get a smooth groupoid that encodes the smooth deformation of the pair $M\times M$ to the tangent group bundle GM. This shows that, at every point, the tangent group is the tangent space in a true differential-geometric fashion. Moreover, the very fact that we have a groupoid accounts for the group structure of the tangent group. Incidentally, this answers a well-known question of Bellaïche [11].     

可极化Carnot群上一类非散度型方程的非平凡解

§1Introduction WebeginbyquotingsomepreliminaryfactsontheCarnotgroupandreferthein estedreaderto[1-3]formorepreciseinformationonthissubject.ALiegroupG=(Rn,o)isaCarnotgroupifthefollowingproperties(G1)(G2)hold.(G1)RncanbesplitintorsubspacesRn=Rn1×...×Rnrandthedilations(δλdefinedbyδλ(x)=(λx(1),λ2x(2),...,λrx(r)),x(j)∈Rnj areautomorphismsofG.(G2)TheLiealgebragofGisgeneratedbytheleftinvariantvectorfieldsX1,.Xn1satisfying Xj(0)=xj,j=1,...,n1.Thenaturalnumbersrand Q=n1+2n2+...+rnr…     

Liouville nonintegrability of sub-Riemannian problems on free Carnot groups of step 4

One of the main approaches to the study of the Carnot–Carathéodory metrics is the Mitchell–Gromov nilpotent approximation theorem, which reduces the consideration of a neighborhood of a regular point to the study of the left-invariant sub-Riemannian problem on the corresponding Carnot group. A detailed analysis of sub-Riemannian extremals is usually based on the explicit integration of the Hamiltonian system of Pontryagin’s maximum principle. In this paper, the Liouville nonintegrability of this system for left-invariant sub-Riemannian problems on free Carnot groups of step 4 and higher is proved.     

Intrinsic Lipschitz Graphs Within Carnot Groups

A Carnot group is a connected, simply connected, nilpotent Lie group with stratified Lie algebra. We study the notions of intrinsic graphs and of intrinsic Lipschitz graphs within Carnot groups. Intrinsic Lipschitz graphs are the natural local analogue inside Carnot groups of Lipschitz submanifolds in Euclidean spaces, where “natural” emphasizes that the notion depends only on the structure of the algebra. Intrinsic Lipschitz graphs unify different alternative approaches through Lipschitz parameterizations or level sets. We provide both geometric and analytic characterizations and a clarifying relation between these graphs and Rumin’s complex of differential forms.     

Almost periodic viscosity solutions of nonlinear evolution equations in Carnot groups

The aim of this paper is to establish existence and uniqueness of time almost periodic viscosity solutions to first-order evolution equations in Carnot groups.     

On unique continuation properties for sub-Laplacian on Carnot groups

In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.     

ON UNIQUE CONTINUATION PROPERTIES FOR THE SUB-LAPLACIAN ON CARNOT GROUPS

In this article, authors begin with establishing representation formulas and properties for functions on Carnot groups. Then, some unique continuation results to solutions of sub-Laplace equations with potentials are proved.