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

三个项目将重点做数学工作 关于非线性偏微分方程理论中存在的问题 微分方程 第一个是关于问题 与超定边值问题中的对称性有关。 对称性出现在某些方程的解中， 正解的存在性意味着该区域是一个 球和解决方案是径向对称的。 工作会完成的 检查积极性假设可能在多大程度上 当人们仍然可以推断出域的对称性时，将其丢弃。 与这项调查有关的是关于平均数的问题 当集合服从刚性约束时， 在空间中移动如果假设平均数是 零，函数必须为零，一个说庞培 财产持有。 决定法律效力的问题 性质等价于证明解的存在性 拉普拉斯算子的特征值问题 工作将在 寻找集合的几何性质， 分析结果。 第二个项目涉及的问题，从潜在的理论 在其中，知识的解决方案的定量性质， 正在寻找相关运营商。 一个特别的问题是， 给定边界上的几何条件的问题 域，其特征在于热的正则点 操作符. 相关工作将考虑边界条件 从这个角度，我们可以测量 解决方案 在第三个项目中，工作将集中在一个新的 椭圆与非椭圆唯一性的探讨 操作员不是基于经典的Carleman方法。 最近的研究集中在包含 无界低阶项 目的是确定何时 开域上等于零的齐次方程的解 set必须处处等于零。 Carleman在1939年的一项研究结果表明， 影响了该领域的所有后续结果。 新发现 将采用几何和变分思想相结合的方法 为了扩展目前的理论，以涵盖更大的类， 运营商

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.