共查询到20条相似文献,搜索用时 0 毫秒
1.
The main result of this article is the existence and uniqueness of the solution of the Dirichlet problem for quaternionic
Monge-Ampère equations in quaternionic strictly pseudoconvex bounded domains in ℍ
n
. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2]. 相似文献
2.
Erhard Heinz Ralf Beyerstedt 《Calculus of Variations and Partial Differential Equations》1994,2(2):241-247
Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt – s
2=E in the punctured disk 0<(x–x
0)2+(y–y
0)2<2. Assume thatq is continuous at (x0, y0). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x
0,y
0) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344). 相似文献
3.
4.
Luis A. Caffarelli & Yu Yuan 《分析论及其应用》2022,38(2):121-127
We construct merely Lipschitz and $C^{1,α}$ with rational $α ∈ (0, 1 − 2/n]$ viscosity solutions to the Monge-Ampère equation with constant right hand side. 相似文献
5.
对一类Monge-Amp(e)re方程的特征值问题进行了研究.通过移动平面法证明了在凸对称区域内,Dirichlet问题的C2凹(凸)解一定是对称的.进而通过对常微分方程和椭圆形偏微分方程的讨论,得到一类n维单位球上特征值问题的非平凡解的存在性和正则性结果. 相似文献
6.
王伟叶 《数学年刊A辑(中文版)》2007,(3)
对一类Monge-Ampère方程的特征值问题进行了研究.通过移动平面法证明了在凸对称区域内,Dirichlet问题的C~2凹(凸)解一定是对称的.进而通过对常微分方程和椭圆形偏微分方程的讨论,得到一类n维单位球上特征值问题的非平凡解的存在性和正则性结果. 相似文献
7.
汪徐家 《数学年刊A辑(中文版)》1992,(1)
本文讨论Monge-Ampere方程的斜微商问题,证明了二维Monge-Ampere方程经典解的存在性以及多维Monge-Ampere方程的广义解的存在性。 相似文献
8.
Yanyan Li & Siyuan Lu 《分析论及其应用》2022,38(2):128-147
We consider the Monge-Ampère equation det $(D^2u) = f$ in $\mathbb{R}^n,$ where $f$ is a
positive bounded periodic function. We prove that $u$ must be the sum of a quadratic
polynomial and a periodic function. For $f ≡ 1,$ this is the classic result by Jörgens, Calabi and Pogorelov. For $f ∈ C^α,$ this was proved by Caffarelli and the first named
author. 相似文献
9.
In this paper, we study the Dirichlet problem for a singular Monge-Amp`ere type equation on unbounded domains. For a few special kinds of unbounded convex domains, we find the explicit formulas of the solutions to the problem. For general unbounded convex domain ?, we prove the existence for solutions to the problem in the space C∞(?) ∩ C(?). We also obtain the local C1/2-estimate up to the ?? and the estimate for the lower bound of the solutions. 相似文献
10.
Alessandro De Paris Alexandre M. Vinogradov 《Central European Journal of Mathematics》2011,9(4):731-751
All second order scalar differential invariants of symplectic hyperbolic and elliptic Monge-Ampère equations with respect
to symplectomorphisms are explicitly computed. In particular, it is shown that the number of independent second order invariants
is equal to 7, in sharp contrast with general Monge-Ampère equations for which this number is equal to 2. We also introduce
a series of invariant differential forms and vector fields which allow us to construct numerous scalar differential invariants
of higher order. The introduced invariants give a solution of the symplectic equivalence of Monge-Ampère equations. As an
example we study equations of the form u
xy
+ f(x, y, u
x
, u
y
) = 0 and in particular find a simple linearization criterion. 相似文献
11.
12.
Ph. Delanoë 《manuscripta mathematica》1983,45(1):29-45
In [4] (Theorem 2), we solved a Monge-Ampère equation, on a compact Riemannian manifold, which isnot a priori locally invertible. In particular, one cannot expect the solution to be unique. We investigate here, the way bifurcation phenomena may occur in such a case, with some explicit computations on flat tori. 相似文献
13.
A. M. Verbovetsky R. Vitolo P. Kersten I. S. Krasil’shchik 《Theoretical and Mathematical Physics》2012,171(2):600-615
We consider a third-order generalized Monge-Ampère equation uyyy ? u xxy 2 + uxxxuxyy = 0, which is closely related to the associativity equation in two-dimensional topological field theory. We describe all integrable structures related to it: Hamiltonian, symplectic, and also recursion operators. We construct infinite hierarchies of symmetries and conservation laws. 相似文献
14.
Robert J. Berman Sébastien Boucksom Vincent Guedj Ahmed Zeriahi 《Publications Mathématiques de L'IHéS》2013,117(1):179-245
We show that degenerate complex Monge-Ampère equations in a big cohomology class of a compact Kähler manifold can be solved using a variational method, without relying on Yau’s theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kähler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kähler metrics and Berndtsson’s positivity of direct images, we extend Ding-Tian’s variational characterization and Bando-Mabuchi’s uniqueness result to singular Kähler-Einstein metrics. Finally, using our variational characterization we prove the existence, uniqueness and convergence as k→∞ of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy. 相似文献
15.
F. Labourie 《Geometric And Functional Analysis》1997,7(3):496-534
Riemann's Uniformization theorem is a classical tool for the study of elliptic problems on surfaces. Usually, the use of
this theorem reflects the fact that the situation can be translated in a pseudo-holomorphic language: the solutions of the
problem appearing as holomorphic curves for a suitable almost complex structure in a jet space. Often, the lack of compactness
of the space of solutions of bounded energy is remarkably described by Gromov's compactness theorem on holomorphic curves.
On the other hand for other problems, usually related to Monge-Ampère equations, a different type of lack of compactness appears;
solutions with bounded energy converge and, furthermore, it is possible to describe what happens when the energy goes to infinity:
the solutions tend to degenerate along holomorphic curves described by solutions of ODE. The goal of this article is to describe
the "Monge-Ampère geometry" of the jet-space that corresponds to this phenonemon. We prove compactness results for the solutions of these problems,
and show examples and applications of our technique. Furthermore, a moduli space of pointed solutions is exhibited with its
structure of a riemaniann lamination.
Submitted: December 1995, revised version: September 1996, final version: March 1997 相似文献
16.
万细仔 《数学年刊A辑(中文版)》1989,(6)
本文给出了方程(A)的解的存在性,唯一性和正则性结果。其中D为R~n中严格凸的,具有D~4光滑边界αD的有界区域,T>0,0<β<1为常数,f(x,t,u,p),F(x,t)为函数,满足: 相似文献
17.
18.
19.
20.
We first define molecules for Hardy spaces
H1F(\mathbbRn)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}) associated with a family F\mathcal{F} of sections which is closely related to the Monge-Ampère equation and prove their molecular characters. As an application,
we show that Monge-Ampère singular operators are bounded on
H1F(\mathbbRn)H^{1}_{\mathcal{F}}(\mathbb{R}^{n}). 相似文献