Mathematical Sciences: "Unique Continuation, Quantitative Properties of Solutions and Symmetry for PDE's

本课题将研究亚椭圆算子的唯一延拓，Wiener准则和Harnack不等式，边值问题的对称性。亚椭圆算子的唯一延拓将通过混合几何和变分技术来研究，这种技术允许比通常的Carleman方法更多的情况。在对称性的研究中，PI将继续尝试解决希弗猜想和庞培问题。在这个项目中要考虑的问题是由几何和数学物理的基本问题激发的。他们的答案将有助于更牢固地理解基本科学问题。

This project will study unique continuation for subelliptic operators, Wiener's criterion and Harnack inequalities, and symmetry of boundary value problems. Unique continuation for subelliptic operators will be studied through a blend of geometric and variational techniques that allow more cases than the usual Carleman method. In the studies of symmetries the PI will continue his attempts in the solution to Schiffer's conjecture and Pompeiu's problem. The questions to be considered in this project are motivated by basic problems in geometry and mathematical physics. Their answers will contribute to a firmer understanding of fundamental scientific issues.