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.

三个项目将集中解决非线性偏微分方程理论中出现的问题。 第一个涉及超定边值问题中与对称性相关的问题。 对称性出现在某些方程的解中，其中正解的存在意味着域是球并且解是径向对称的。 我们将开展工作，检查在仍然可以推断域对称性的情况下，积极性假设可能被放弃的程度。 与这项研究相关的是关于固定集合上函数的平均值的问题，因为该集合会在空间中进行刚性运动。如果假设平均值为零，则函数必定为零，则可以说庞贝性质成立。 判断属性有效性的问题相当于证明拉普拉斯特征值问题解的存在性。 我们将致力于寻找补充该分析结果的集合的几何属性。 第二个项目涉及势理论中的问题，其中寻求相关算子解的定量属性的知识。 一个特殊的问题是在域边界上给出几何条件的问题，该几何条件表征热算子的规则点。 相关工作将考虑边界条件，从中可以测量解的非切向极限的范围。 在第三个项目中，工作将集中于一种不基于经典卡尔曼方法的椭圆和非椭圆算子唯一性属性的新方法。 最近的研究集中在包含无界低阶项的算子上。 目的是确定在开集上为零的齐次方程的解何时必须处处为零。 卡尔曼 1939 年的一项结果影响了该领域的所有后续结果。 结合几何和变分思想的新发现将被用来扩展当前的理论以涵盖更大的算子类别。