Mathematical Sciences: Harmonic Analysis and the Laplace Equation

该奖项支持的工作将集中在 以椭圆偏微分方程为中心的数学问题 微分方程 主题是独特的 继续解决问题和相关问题。 由唯一 继续一个理解的问题，表明有 是没有（非零）的解决方案消失在一个开放的集合。 第二 问题，其中一个问是否解决方案可以消失， 在一点上的无穷级称为强连续 财产 两者都是关于两个人是否 在一个开集（无限阶， （1）要永远保持一致。一些积极的结果是 已知，这取决于方程的平滑度和 基本维度。 最近的例子表明， 某些经典方程，薛定谔与加藤类 潜在的，唯一的延续失败。 具体目标之一 这个项目的目的是扩展最好的基础工具 这就是所谓的Carleman不等式。这些不平等 只适用于常系数方程。进一步分析 更普遍的不平等表明，在以下方面可以取得进展： 建立良好的估计，可适用于一般 椭圆方程 相关工作将在Bers上进行 一个问题，它问的是梯度为零的解是否为 边界处的某个度数必须降为常数。 抽象的，因为这些问题出现在那里一般的形式， 他们很容易被认为是重要的观点， 数学的使用者，特别是在那些情况下， 解的近似可能导致伪解。 知识 唯一性的延续可以提供一定程度的信心 合理的近似会导致正确的答案。

Work to be supported by this award will focus on Mathematical questions centering on elliptic partial differential equations. The theme is that of unique continuation of solutions and related issues. By unique continuation one understands the problem of showing that there are no (non-zero) solutions vanishing in an open set. A second question, whereby one asks whether a solution can vanish to infinite order at a point is called the strong continuation property. Both are variations on the question of whether two solutions which agree on an open set (of infinite order at a point) must agree forever. A number of positive results are known, depending both on the smoothness of the equation and the underlying dimension. Recent examples show that in the case of certain classical equations, Schrodinger with Kato class potentials, unique continuation fails. One of the specific goals of this project is to extend the basic tool underlying the best results, known as the Carleman inequalities. These inequalities only apply to constant-coefficient equations. Further analysis of more general inequalities suggest that progress can be made in establishing good estimates which can be applied to general elliptic equations. Related work will be carried out on the Bers problem which asks whether solutions whose gradient vanish to a certain degree at the boundary must reduce to constants. Abstract as these problems appear in there generic form, they are easily seen to be important from the point of view of users of mathematics, especially in those cases where approximation of solutions can lead to spurious ones. Knowledge of uniqueness of continuation can provide a level of confidence that reasonable approximations lead to the correct answer.