椭圆和抛物方程解的边界正则性

In this program, we study the boundary regularity of solutions of elliptic and parabolic equations. It is well known that the geometric properties of domains have significant influence on the boundary regularity of solutions. There are many remarkable results in this respect. In recent years, it causes lots of research interest to obtain the a priori estimates under weak hypothesis on the coefficients or the geometric properties of the boundary. In this program, we intend to obtain the following three results focusing on the boundary regularity of elliptic and parabolic equations:.(1).C^{1,alpha} and C^{1,Dini} estimates for strong solutions of elliptic (parabolic) equations in nondivergence form under weak hypothesis on the lower order terms of the equations..(2).Interior and boundary Harnack inequalities for parabolic equations in nondivergence form with unbounded lower order term..(3).Boundary differentiability for solutions of elliptic equations in divergence form on convex domains.

本项目研究椭圆和抛物方程解的边界正则性问题。众所周知，区域的几何性质对解的边界正则性有至关重要的影响。在这方面，历史上有很多著名的结果。近几年，减弱系数或边界几何假设条件来得到先验估计正逐渐成为椭圆与抛物方程领域的一个研究热点。本项目拟研究散度和非散度椭圆和抛物方程边界正则性方面的三个问题：（1）低阶项光滑性假设减弱时，研究非散度型椭圆(抛物)方程的解在非平坦边界的C^{1,alpha},C^{1,Dini}估计，并研究凸区域上的边界可微性；（2）含无界低阶项的非散度型抛物方程的内部Harnack不等式和边界Harnack不等式。 （3）凸区域上的散度型椭圆方程解的边界可微性；

本项目研究椭圆和抛物方程解的边界正则性问题。区域的几何性质对解的边界正则性有至关重要的影响。本项目在执行过程中，我们主要研究了： (1) 当区域边界满足某种最优条件时非散度型椭圆方程解的边界可微性；（2）对一类退化椭圆方程，我们证明了解的边界C^{1,\alpha}正则性; (3)在低阶项系数b在临界空间L^n的情形，证明了非齐次右端项情形非负解的内部Harnack不等式； 之后当b的L^n模满足Dini条件，区域为C^{1,Dini}或者Reifenberg C^{1,Dini} 时，证明了解在边界的可微性以及相应的估计；再进一步将ReifenbergC^{1,Dini}的定义进行推广，考虑边界满足外部ReifenbergC^{1,Dini}且是Reifenberg C^{1}，可以得到解边界的可微性；相应的结果对抛物方程情形也是成立的。