非线性椭圆型偏微分方程的边界正则性

项目名称： 非线性椭圆型偏微分方程的边界正则性

项目编号： No.11201250

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 马飞遥

作者单位： 宁波大学

项目金额： 22万元

中文摘要： 本项目将研究非线性椭圆型偏微分方程的边界正则性，旨在探索解的正则性与边界的正则性、边界值以及扰动项之间的最佳估计。我们考虑完全非线性椭圆方程Dirichlet边值问题以及Oblique边值问题粘性解的正则性。已有的边界正则性研究结果对边界的光滑性要求高，本研究的目标是要降低边界光滑性假设并得到一些新的正则性估计。将得到新的闸函数，优化的迭代和新的估计，包括Alexander-Backelman-Pucci型估计、边界Harnack不等式和各阶导数的连续模估计。还将研究此类估计在Monge-Ampere方程和完全非线性抛物型方程中的推广。多项式逼近、De Giorgi方法、紧方法和申请人的边界正则性理论中的新方法等都将是本课题的重要研究工具。本项目的研究内容是偏微分方程的基本问题，研究结果将促进人们对椭圆型偏微分方程边界正则性的深入理解，一些结果也能用于自由边界、激波和应用领域中的相关问题。

中文关键词： 正则性；椭圆型偏微分方程；完全非线性问题；斜导数问题；非光滑边界

英文摘要： In this project we will study the boundary regularity of nonlinear elliptic partial differential equations,in particular the relation between the regularity of the solution and the optimal hypotheses for smoothness of domain, boundary value condition and perturbation. We consider the viscosity solutions of the Dirichlet boundary value problem and oblique boundary value problem of the fully nonlinear elliptic equations. The known results of the regularity theory for these equations need sufficient smoothness of the domain. Our main aim is to investigate the problem weakening the regularity assumptions on the smoothness of the domain. We will obtain new barrier functions, optimal iteration and estimates including Alexander-Backelman-Pucci type estimate,boundary Harnack inequality and the modulus of continuity estimates of the viscosity solutions and their derivatives. Furthermore, we shall explore the generalization of such theorems to the Monge-Ampere equations and fully nonlinear parabolic equations. Many skills will be useful for our study, which include polynomial approximation, De Giorgi's method, compact method and the applicant's new methods for boundary regularity theory. Our study will be helpful to understand the boundary regularity of elliptic equations. It is expected to have applications in the f

英文关键词： Regularity；Elliptic partial differential equation；Fully nonlinear problem；Oblique derivative problem；Nonsmooth boundary