Carleman estimates with nonconvex weights and Riesz rearrangement inequalities

DMS 0407090Alexandru Ionescu威斯康星大学Carleman估计与非凸权重研究将集中在两个主要方向。第一个目标是了解长程扰动在欧氏空间上Schr dinger算子的Carleman不等式中的作用.这些不等式对Schr dinger方程解的唯一性问题有应用.第二个目标是进一步发展可用于半单李群和对称空间分析各个方面的实变量方法。特别是，PI对理解高真实的秩半单李群上的Riesz重排不等式感兴趣。PI建议研究的许多类型的偏微分方程起源于物理，化学或工程。这些方程的作用是模拟某些现象，它们的相关性往往是数字验证。PI建议严格研究这些方程，并确认解的预期行为，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.