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Extremisers and near-extremisers for central inequalities in harmonic analysis, geometric analysis and PDE

调和分析、几何分析和偏微分方程中中心不等式的极值和近极值

基本信息

批准号：
EP/J021490/1
负责人：
Susana Gutierez
金额：
$ 12.23万
依托单位：
University of Birmingham
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FJ021490%2F1
关键词：
Extremisers near extremisers central inequalities

项目摘要

Isoperimetric inequalities are examples of sharp inequalities with intrinsic geometric content. The classical sharp inequality in two dimensions compares the area of an enclosed region in the plane with the length of its perimeter. Thanks to the sharpness, it is possible to deduce immediately that amongst all regions with a fixed perimeter length, the circle produces the maximum area. More generally, isoperimetric inequalities of this flavour can be used to identify maximising or minimising objects within a class, and they have applications in physical problems (for example, liquid droplet formation). Furthermore, these physical applications often demand that a "stable" version of the underlying sharp inequality is proved (to allow for the effect of external perturbative forces, for instance).The main thrust of this broad project is to understand the sharpness and stability of some central inequalities which occur in harmonic analysis, geometric analysis and differential equations. These include the powerful Brascamp-Lieb inequality, which can be viewed as a generalised isoperimetric-type inequality and has fascinating and wide-ranging applications in the above fields and beyond. The project also encompasses sharp space-time inequalities for solutions of important differential equations, including the Schrodinger and wave equations.

等周不等式是具有内在几何内容的尖锐不等式的例子。二维中的经典锐不等式将平面中封闭区域的面积与其周长进行比较。由于其清晰度，可以立即推断出在具有固定周长的所有区域中，圆产生最大面积。更一般地，这种风格的等周不等式可用于识别类内对象的最大化或最小化，并且它们在物理问题（例如，液滴形成）中具有应用。此外，这些物理应用通常要求证明潜在的尖锐不等式的“稳定”版本（例如，考虑到外部扰动力的影响）。这个广泛项目的主要目标是了解调和分析、几何分析和微分方程中出现的一些中心不等式的尖锐性和稳定性。其中包括强大的 Brascamp-Lieb 不等式，它可以被视为广义的等周型不等式，并且在上述领域及其他领域具有令人着迷且广泛的应用。该项目还包含重要微分方程解的尖锐时空不等式，包括薛定谔方程和波动方程。