Carleman estimates with nonconvex weights and Riesz rearrangement inequalities

DMS 0407090威斯康星大学的Alexandru Iones cube估计非凸权重Carleman估计本研究将集中在两个主要方向。第一个目的是了解长程扰动在欧氏空间上Schr‘o dinger算子的某些Carleman不等式中的作用.这些不等式可应用于Schr’o dinger方程解的唯一性问题.第二个目标是进一步发展可用于分析半单李群和对称空间的各个方面的实变量方法。特别是，PI对高实数阶半单李群上的Riesz重排不等式的理解很感兴趣。PI建议研究的许多类型的偏微分方程组都起源于物理、化学或工程。这些方程的作用是模拟某些现象，它们的相关性通常是通过数字来验证的。PI建议严格研究这些方程，并确定解的预期行为，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。

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.