Carleman estimates with nonconvex weights and Riesz rearrangement inequalities

使用非凸权重和 Riesz 重排不等式进行 Carleman 估计

• 批准号: 0407090
• 负责人: Alexandru Ionescu
• 金额: $ 11.06万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0407090&HistoricalAwards=false
• 关键词: Carleman; estimates; nonconvex; weights; Riesz

DMS 0407090Alexandru IonescuUniversity of WisconsinCarleman estimates with nonconvex weightsThe research will focus in two main directions. The first objective is to understand the role of long-range perturbations in certain Carleman inequalities for Schr\"{o}dinger operators on Euclidean spaces. Theseinequalities have applications to questions concerning uniqueness of solutions of Schr\"{o}dinger equations. The second objective is to further develop real-variable methods that can be used in various aspects of analysis on semisimple Lie groups and symmetric spaces. In particular,the PI is interested in understanding Riesz rearrangement inequalities on semisimple Lie groups of high real rank.Many partial differential equations of the type the PI proposes to study originate from physics, chemistry or engineering. The role of these equations is to model certain phenomena, and their relevance is often verified numerically. The PI proposes to study these equations rigorously, and confirm the expected behavior of solutions, such as the infinite speed of propagation of solutions of large classes of nonlinear Schr\"{o}dinger equations, as mathematical theorems.

DMS 0407090Alexandru Ionescu 威斯康星大学卡尔曼利用非凸权重进行估计 该研究将集中在两个主要方向。第一个目标是了解欧几里得空间上薛定谔算子的某些卡尔曼不等式中长程扰动的作用。这些不等式可应用于有关薛定谔方程解的唯一性的问题。第二个目标是进一步开发实变量方法，可用于半单李群和对称空间分析的各个方面。特别是，PI 对理解高实秩半单李群上的 Riesz 重排不等式很感兴趣。PI 建议研究的许多偏微分方程类型都源自物理、化学或工程学。这些方程的作用是对某些现象进行建模，并且它们的相关性通常经过数值验证。 PI 建议严格研究这些方程，并确认解的预期行为，例如大类非线性薛定谔方程解的无限传播速度，作为数学定理。