Mellin Transform and Global Optimization Techniques for Partial Differential Equations

摘要：非光滑域上椭圆边值问题的研究是调和分析与偏微分方程（PDEs）、泛函分析与算子理论相结合的中心问题。在欧几里得环境下求解数学物理中许多经典偏微分方程的一种典型方法是层势法。PI计划解决一些重要的和长期存在的问题，这些问题是关于利普希茨域上边界层势算子的基本性质的，尽管在过去的三十年里取得了巨大的进展，但这些问题仍然是开放的。该研究计划的一些具体主题是粗糙域上边界层的谱半径猜想；粗糙边界多连通区域弹性位移规定问题格林函数和泊松核的正则性。对具有最小平滑假设的边值问题（BVPs）的研究（在所讨论的系数或域上）一直是偏微分方程领域基本发展背后的驱动力，并与分析的许多领域有着深刻的联系。对这些问题的兴趣既在于它们在理论上的重要性，也在于它们在工程上的大量应用。

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.