非线性椭圆型偏微分方程的边界正则性

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 fields of free boundary, shock wave and some other related problems.

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。

本项目围绕非线性椭圆方程的边界正则性进行研究。首先对完全非线性一致椭圆方程的Dirichlet边值问题得到边界一阶导数Holder正则性的新证明（用紧方法结合边界逼近）。对完全非线性一致抛物型方程的的Dirichlet边值问题得到了Lateral边界的解的可微性及一阶导数的连续模估计。本项目也对完全非线性一致椭圆方程的Oblique边值问题得到了Oblique条件下粘性解的A-B-P型极值原理，得到了Lipschitz边界的Holder估计，也研究了一般区域边界的一阶导数二阶导数Holder正则性。此外研究了退化的p拉普拉斯型Baouendi–Grushin方程，得到了其弱解的q次可积性估计。还得到了一类双曲型曲率流方程解的局部存在性和保凸性。此外还研究了一类Novikov型方程的持续性质。此项目还有很多值得研究的相关问题，我们将继续研究下去。