Mathematical Sciences: Gain of Regularity

数学科学：规律性的增益

• 批准号: 9002152
• 负责人: Thomas Kappeler
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-04-15 至 1992-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9002152&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Gain; Regularity

One central theme of this project focuses on nonlinear partial differential equations representing the evolution of certain physical phenomena. An evolution equation whose solutions are smoother than the initial data is said to have a gain of regularity. An example of this occurs in the Schrodinger equation if the initial data decay faster than any polynomial. Some success has also been obtained in the nonlinear area, particularly in the case of the Korteweg-de Vries equation. In this work, a general class of equations will be investigated for gain in regularity. It will concentrate on equations in three space dimensions and seek to prove that gain in regularity is governed by the sign of the third derivative of the solution. Method of proof uses the technique of nonlinear multipliers and a fundamental energy (integral) estimate which shows the gain. Efforts will also be made in applying similar ideas to dispersive equations such as nonlinear Schrodinger equations. A second line of investigation concerns spectral and inverse spectral problems of the Laplace operator or the Laplace operator plus a potential defined on compact Riemannian manifolds. Three objectives of this work are to: (1) describe the distribution of the eigenvalues, (2) identify the properties of a Riemannian metric or a potential which are spectral invariants, and (3) determine the metrics or potentials which can be isospectrally deformed in a mathematically significant manner. Particular emphasis will be placed on measuring the pattern of gaps between eigenvalues and the density properties of potentials with a finite number of double eigenvalues.

该项目的一个中心主题是非线性 偏微分方程表示的演变 某些物理现象。 一个演化方程， 解决方案比初始数据更平滑， 获得规律性。 这方面的一个例子发生在薛定谔 如果初始数据比任何多项式衰减得更快，则可以使用等式。 在非线性领域也取得了一些成功， 特别是在Korteweg-de弗里斯方程的情况下。 在 这项工作，一般类方程将被调查， 规律性的增益。 它将集中在三个方程 空间维度，并试图证明规律性增益是 由解的三阶导数的符号决定。 证明方法使用非线性乘法器和 基本能量（积分）估计，其显示增益。 还将努力将类似的想法应用于分散的 非线性薛定谔方程。 第二条调查线涉及频谱和逆 拉普拉斯算子或拉普拉斯算子的谱问题 加上一个定义在紧致黎曼流形上的势 三 本研究的目的是：（1）描述 特征值，（2）识别Riemannian的性质 度量或潜在的谱不变量，和（3） 确定可以等谱地 以数学上显著的方式变形。 特别 重点将放在衡量差距之间的模式， 特征值和密度性质 双特征值的有限个数