Harmonic analysis and partial differential equations: Sharp geometric inequalities, fully nonlinear equations and applications

建议的研究项目涉及使用调和分析和偏微分方程的主题。我们将着重于寻找Heisenberg群、分层群、复球面和更一般的CR流形上的Moser-Trudinger-Onofri不等式的尖锐常数和极值，并导出Heisenberg群和更一般的分层群上的尖锐Sobolev不等式，以及高阶微分算子的最优几何不等式。该项目将使用这些研究CR几何中的应用，包括退化椭圆Monge-Ampere方程的正则性，以及高斯曲率流产生的抛物Monge-Ampere方程和次椭圆设置中完全非线性方程的正则性。我们还将研究凸函数在这种情况下的性质和应用，一些几何嵌入不等式，如Poincare和Sobolev不等式与高阶次椭圆导数及其应用偏微分方程。成功完成亚椭圆结构的项目需要大量的新技术和创新思想，这些技术和思想在经典的情况下是不可用的。这些项目在控制理论、优化、人类视觉和其他科学和工程领域都有应用。

The proposed research projects involves topics that use harmonic analysis and partial differential equations. We will focus on finding the sharp constants and extremals for Moser-Trudinger-Onofri inequalities on the Heisenberg group, stratified groups, the complex sphere and more general CR manifolds Also to derive sharp Sobolev inequalities on the Heisenberg and more general stratified groups, and optimal geometric inequalities for high order differential operators. The project will use these to study applications in CR geometry, including the regularity of degenerate elliptic Monge-Ampere equations and also parabolic Monge-Ampere equations arising from the Gauss curvature flow and the regularity of fully nonlinear equations in the sub-elliptic setting. We shall also study the properties and applications of convex functions in such settings, some geometric embedding inequalities such as the Poincare and Sobolev inequalities associated with high order sub-elliptic derivatives and their applications to partial differential equations. The successful completion of the proposed projects in sub-elliptic structures requires substantially new techniques and innovative ideas which are not available in the classical cases.The proposed projects have applications to control theory, optimization, human vision and other topics in sciences and engineering.