分数阶抛物p-Laplace方程解的全局正则性

Recently, fractional partial differential equations have been applied to all spheres of natural science. This program aims at study on global regularity for solutions of fractional parabolic p-Laplace equations. These results are of great academic value in further development of the theory of partial differential equations. The program plays an significant role in guiding such fields as stochastic partial differential equations, image processing and financial options. . This program takes the following four core questions related to fractional parabolic p-Laplace equations as its initial focus. .1. Study ABP Maximum Principle.. 2. Bring in fraction space and study the interior regularity by global ABP Maximum Principle. This program mainly rely on the methods of Compactness, Vitali coverage and C-Z decompositon applicable to fraction equations..3. Construct appropriate bar functions and conduct research on regular effects exerted over solutions by the shape of the boundary and the functions on the boundary.. All these research will contribute to the further probing into full nonlinear fractional parabolic equations and fractional Hamilton-Jacobi equation by offering theoretic experience.

近几年分数阶偏微分方程已经开始应用到自然科学的各个领域。本项目旨在研究分数阶抛物p-Laplace方程解的全局正则性理论。这些结论对发展和完善偏微分方程理论有非常重要的科研价值。本项目的研究对随机偏微分、图像处理、金融期权等领域都有重要的指导作用。. 本项目拟研究分数阶抛物p-Laplace方程中的几个核心问题：一、证明ABP极值原理。二、引入分数阶空间，通过全局ABP极值原理研究方程解的内部正则性估计。适用于分数阶方程的紧方法、Vitali覆盖、C-Z分解等将是证明的重要工具。三、构造分数阶障碍函数，研究边界的几何形状、边界函数对解的正则性影响。. 这些研究将为进一步探索完全非线性分数阶抛物方程、分数阶Hamilton-Jacobi方程的理论积累经验。

分数阶p-Laplace方程是经典p-Laplace分方程的一个自然推广，它是偏微分方程领域近两年刚兴起且最活跃的研究方向之一。其研究对随机偏微分、图像处理、金融数学、期权股权都有重要的推动作用。本项目研究偏微分方程中的几个重要的问题。.1． 首先我们研究了一类高阶加权偏微分方程组，得到了其解的存在性、临界指标和特殊情况下解的对称性。 .2. 我们研究了p-Laplace加权偏微分方程，得到了解的存在性，以及临界指标、临界边值。.3.我们研究了几类p-Laplace型Baouendi-Grushin 方程，得到了其Pohozaev恒等式、解的存在性以及其正则性结论。.4. 我们研究了一类分数阶的Wolff位势积分方程组解的性质。得到了其解的存在性和对称性。.5.我们研究了一类分数阶积分微分方程，得到了其边界正则性估计。