一类分数阶退化椭圆方程的自由边界问题

The aim of this program is to investigate a class of free boundary problems involving fractional order degenerate elliptic equations. The fractional free boundary problems have been wildly applied in such fields as financial markets, viscous fluid dynamics, medical ultrasound and soft matter. .We plan to study some properties of its solution and the free boundary of our fractional free boundary problem in this program, including the existence, the regularity and the increase rate near the free boundary of the solution; the Hausdorff measure, the regularity and the singular structure of the free boundary..In this program, we study not only the regularity of solutions of the free boundary problems involving fractional order degenerate elliptic equations, but also the relation between fractional order free boundary problems and the second order free boundary problems. We hope to understand and solve these problems in a new perspective so as to establish a new comprehensive unified method and theory. This has important significance for the development and improvement of the theory of partial differential equations.

本项目旨在研究一类分数阶退化椭圆方程的自由边界问题。分数阶自由边界问题在金融市场、粘性流体力学、医学超声检测以及软物质等领域有广泛的应用。. 本项目拟得到一类分数阶退化椭圆方程的自由边界问题的解以及自由边界的相关性质，包括解 的存在性、正则性及其在自由边界附近的增长速率，自由边界的Hausdorff测度、正则性以及奇异性结构。. 本项目不只停留在对分数阶退化方程型的自由边界问题本身的正则性研究上，而是进一步探索它与二阶偏微分方程自由边界问题的内在联系，在一个新的高度上以新的视角来认识问题、解决问题，从而建立起新的全面统一的方法和理论。这对整个偏微分方程理论的发展与完善都有很重要的意义。

本项目主要是关于分数阶p-Laplace方程在自由边界问题上的一系列研究。主要研究内容和结果如下：.1.研究了一类特殊的p-Laplace方程，即Baouendi-Grushin算子的p-Laplace方程解的性质，包括正则性估计、对称性估计。.2.得到了一类分数阶Laplace方程在超定问题下边界的性质及方程解的性质。.3.给出了一类分数阶p-Laplace方程在超定问题下边界的性质和方程解的性质。