一些偏微分方程和方程组的定性研究

We plans to study the qualitative properties of entire solutions of elliptic nonlinear partial differential equations and systems. In particular, we will investigate saddle solutions and traveling wave solutions of Allen-Cahn type equations, including the classic scalar equations with a double well potentials and vector-valued equations with multiple well potential. For the stationary scalar Allen-Cahn equation, we will focus on the existence of special saddle solutions with prescribed level sets as well as on the level set structure of solutions of finite Morse index, in particular, on the relation between the level sets of solutions and minimal surfaces. The ultimate goal is to classify all entire solutions of Allen-Cahn equation.For the traveling wave solutions of the Allen-Cahn equation, we will study the symmetry and classification of all monotone solutions and their relations with mean curvature solitons.For the vector-valued equations with triple well potentials, we plans to study the singular sets of certain minimizing solutions with proper boundary conditions and their relations to the triple junction solutions in the entire plane. We will also investigate the axial symmetry and non degeneracy of traveling wave solutions of Gross-Pitaevskii equation. Finally, we will study the existence of solutions of fractional Laplacian in whole Space. All of the above issues are paid close attention because of their physics backgroud.

我们计划研究非线性偏微分方程组或者系统的整体解的适定性。特别我们将研究Allen-Cahn 型方程的鞍形解和行波解，包括具有双井位势的经典标量方程以及具有多重井位势的矢量方程。 对于稳态的标量Allen-Cahn 方程，我们将重点研究特殊鞍解的存在性以及解水平集与极小曲面之间的关系，从而能够将Allen-Cahn方程的所有整体解分类。关于Allen-Chan 方程的行波解，我们将研究所有单调解的对称性和分类以及与平均曲率界之间的关系。关于具有三井位势的矢量方程，我们将研究具有合适边界条件的某特定极小化解的奇异集一以及在整个平面上此奇异集与三结点解之间的关系。我们还将研究Gross-Piaevskii方程的非退化的轴对称行波解。 最后我们还将研究分数阶Laplacian方程在全空间上解的存在性。以上所研究的问题因其强烈的物理背景备受关注。

著名的推广了的变分法被用来构造Allen-Cahn方程在二维欧氏空间的四终端解。当结点处的角度接近pi/2或者0时，利用Lyapunov–Schmid 约化法， Del Pino, Kowalczyk, Pacard 和 Wei构造了四终端解。对于结点处的角度在0和pi/2之间时，Kowalczyk, Liu 和 Pacard 利用连续方法构造了四终端解。在具有一定限制的空间上，通过特殊的山路论证，或者称为单调函数的单调路径的集合，首先通过对节点集很好的控制下构造有界区域上的一族解，然后令区域趋向于整个区域，从而得到四终端解。通过这个方法，不仅仅构造了在角度为在0和pi/2之间的值时的一族四终端解，而且这类四终端解的Morse指标为1也被证明了。