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

数学科学：偏微分方程的对称性、偏微分方程解的定量性质以及唯一连续性

• 批准号: 8905338
• 负责人: Nicola Garofalo
• 金额: $ 1.46万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1989
• 资助国家: 美国
• 起止时间: 1989-06-01 至 1989-09-01
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8905338&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Symmetry; PDE; Quantitative

Three projects will be the focus of mathematical work done on problems arising in the theory of nonlinear partial differential equations. The first is concerned with questions related to symmetry in overdetermined boundary value problems. The symmetry occurs in solutions of certain equations in which the existence of positive solutions implies that the domain is a ball and the solution is radially symmetric. Work will be done examining the degree to which the positivity assumption may be dropped while one can still infer symmetry of the domain. Related to this investigation are questions concerning averages of functions over a fixed set as the set is subject to rigid motions through space. If, on assuming that the averages are zero, the function must be zero, one says that the Pompeiu property holds. The problem of deciding the validity of the property is equivalent to showing the existence of solutions of the eigenvalue problem for the Laplacian. Work will be done in looking for geometric properties of sets which complement this analytic result. The second project concerns questions from potential theory in which knowledge of quantitative properties of solutions of the relevant operator are sought. One particular issue is the problem of giving geometric conditions on the boundary of a domain which characterize the regular points for the heat operator. Related work will consider conditions on the boundary from which one may measure the extent of nontangential limits of solutions. In the third project, work will concentrate on a new approach to uniqueness properties of elliptic and non-elliptic operators that is not based on the classical Carleman method. Recent studies have concentrated on operators containing unbounded lower order terms. The object is to determine when solutions of the homogeneous equation which equal zero on an open set must equal zero everywhere. A 1939 result of Carleman has influenced all subsequent results in this area. New discoveries using a blend of geometric and variational ideas will be employed to extend the present theory to cover larger classes of operators.

三个项目将是在非线性部分微分方程理论中出现的数学工作的重点。 第一个关注与超确定边界价值问题中的对称性有关的问题。 对称性发生在某些方程式的溶液中，在某些方程式中，阳性溶液的存在意味着域是一个球，溶液在径向对称。 将完成检查可以删除阳性假设的程度，而人们仍然可以推断域的对称性。 与此调查相关的是关于固定集合的功能平均值的问题，因为该集合可能会通过空间进行严格的运动。如果假设平均值为零，则该函数必须为零，则说庞贝属属性属性。 确定财产有效性的问题等同于显示拉普拉斯（Laplacian）特征值问题的解决方案的存在。 将在寻找补充该分析结果的集合的几何特性时完成的工作。 第二个项目涉及潜在理论的问题，其中寻求有关解决方案解决方案的定量特性的知识。 一个特殊的问题是在域的边界上提供几何条件的问题，该域的特征是热运算符的常规点。 相关工作将考虑在边界上可以衡量解决方案限制程度的边界上的条件。 在第三个项目中，工作将集中在不基于经典卡尔曼方法的椭圆形和非椭圆运算符的独特性能的新方法上。 最近的研究集中在包含无界较低级项的操作员上。 该对象是确定均匀方程的解何时在开放集上等于零的均匀方程式必须等于零。 1939年，卡尔曼（Carleman）的结果影响了该领域的所有后续结果。 使用几何和各种思想的混合物的新发现将被采用以扩展当前的理论，以覆盖更大的运营商。