Mathematical Sciences: Harmonic Analysis and the Laplace Equation

数学科学：调和分析和拉普拉斯方程

• 批准号: 9307872
• 负责人: Thomas Wolff
• 金额: $ 18万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-05-15 至 1996-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9307872&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Harmonic; Analysis; Laplace

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.

该奖项支持的工作将集中在围绕椭圆型偏微分方程的数学问题上。主题是解决办法和相关问题的独特延续。通过唯一的延拓，人们理解了在开集上不存在消失的(非零)解的问题。第二个问题是强连续性质，通过这个问题，一个解是否可以在某一点消失到无限阶。这两个问题都是关于两个在开集上一致的解(在一点上是无穷级的)是否必须永远一致的问题的变体。已知许多积极的结果，这取决于方程的光滑性和潜在的维度。最近的例子表明，对于某些经典方程，具有Kato类位势的薛定谔方程，唯一延拓是失败的。该项目的具体目标之一是扩展作为最佳结果基础的基本工具，即众所周知的Carleman不等。这些不等式只适用于常系数方程。对更一般的不等式的进一步分析表明，在建立可应用于一般椭圆型方程的良好估计方面可以取得进展。将对Bers问题进行相关工作，该问题询问在边界处梯度消失到一定程度的解是否必须降为常量。由于这些问题是以一般形式出现的，从数学用户的角度来看，它们很容易被认为是重要的，特别是在解的近似可能导致虚假解的情况下。关于延拓唯一性的知识可以提供一定程度的信心，即合理的近似会导致正确的答案。