Mathematical Sciences: Gain of Regularity

这个项目的一个中心主题集中在代表某些物理现象演变的非线性偏微分方程上。如果一个演化方程的解比初始数据更平滑，我们就说它具有规律性的增益。在薛定谔方程中，如果初始数据衰减速度比任何多项式都快，就会出现这种情况。在非线性领域也取得了一些成功，特别是在Korteweg-de Vries方程的情况下。在这项工作中，将研究一类一般方程的正则性增益。它将集中于三维空间中的方程，并试图证明正则性的增益是由解的三阶导数的符号决定的。证明方法采用非线性乘法器技术和显示增益的基本能量（积分）估计。我们还将努力将类似的思想应用于色散方程，如非线性薛定谔方程。第二条研究线涉及拉普拉斯算子或拉普拉斯算子加紧黎曼流形上定义的势的谱和逆谱问题。这项工作的三个目标是：(1)描述特征值的分布，(2)确定谱不变量的黎曼度规或势的性质，以及(3)确定可以以数学上有意义的方式进行等谱变形的度规或势。特别强调的是测量具有有限双特征值的势的密度特性和特征值之间的间隙模式。

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.