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上。主题是解决方案和有关问题的独特延续。通过唯一延拓，我们可以理解证明在开集中不存在（非零）解消失的问题。第二个问题是，一个解能否在某一点上消失到无限阶，这个问题被称为强延拓性。两者都是关于两个解是否在一个开集上一致（在一点上有无限阶）必须永远一致的问题的变体。已知的肯定结果有很多，这取决于方程的平滑度和基本维度。最近的例子表明，对于某些经典方程，具有加藤类势的薛定谔方程，唯一延拓是不成立的。该项目的一个具体目标是扩展最佳结果的基本工具，即Carleman不等式。这些不等式只适用于常系数方程。对更一般不等式的进一步分析表明，可以在建立适用于一般椭圆方程的良好估计方面取得进展。在Bers问题上也将进行相关的工作，该问题是关于在边界处梯度消失到一定程度的解是否必须化简为常数的问题。虽然这些问题以抽象的一般形式出现，但从数学使用者的角度来看，它们很容易被认为是重要的，特别是在近似解可能导致虚假解的情况下。延拓的唯一性知识可以提供一定程度的信心，使合理的近似可以得到正确的答案。

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.