Mellin Transform and Global Optimization Techniques for Partial Differential Equations

主要研究者：Irina Mitrea，Cornell UniversityDMS-0245466摘要：一方面，调和分析和偏微分方程（PDE）之间的接口，以及泛函分析和算子理论，另一方面，一个中心问题是研究非光滑域上的椭圆边值问题。层势法是求解欧氏空间Lipschitz域上数学物理中许多经典偏微分方程的一种典型方法。PI计划解决一些重要和长期存在的问题，这些问题涉及Lipschitz域上边界层势算子的基本性质，尽管在过去三十年中取得了巨大进展，但这些问题仍然是开放的。一些具体的主题，这一拟议的研究计划是粗糙域上的边界层的谱半径猜想;位移规定的问题，弹性在多连通区域与粗糙边界;绿色函数和Poisson核的正则性.具有极小光滑性假设的边值问题的研究（关于系数或讨论中的域）一直是偏微分方程领域基本发展背后的驱动力，并与许多分析领域有着深刻的联系。对这些问题的兴趣既在于它们的理论重要性，也在于它们在工程中的众多应用。

PI: Irina Mitrea, Cornell UniversityDMS-0245466Abstract:A central problem at the interface between harmonic analysis and partial differential equations (PDEs), on the one hand, and functional analysis and operator theory, on the other hand, is the study of elliptic boundary value problems on non-smooth domains. One typical approach for solving many classical PDEs of mathematical physics in Lipschitz domains in the Euclidean setting is the method of layer potentials. The PI plans to address a number of important and long-standing questions regarding fundamental properties of boundary layer potential operators on Lipschitz domains, which still remain open despite the tremendous progress made in the last three decades. Some of the specific themes of this proposed research program are The Spectral Radius Conjecture for boundary layers on rough domains;Displacement prescribed problems of elasticity in multi-connected regions with rough boundaries; and Regularity properties of Green functions and Poisson kernels.The study of boundary value problems (BVPs) with minimal smoothness assumptions (on the coefficients or on the domain under discussion) has been the driving force behind fundamental developments in the area of Partial Differential Equations and has deep connections with many areas of Analysis. The interest in such problems resides both in their theoretical importance as well as in their numerous applications to engineering.