一类完全非线性积分-微分椭圆方程研究

The partial differential equations (PDEs) in probability theory are not only nonlinear,but also non-local. Due to the importance in stochastic control and mathematical challenges, these PDEs have been central issues in PDEs. In this proposal, we plan to adopt the classical theory and some new techniques, which have been developed in the studies of non-local PDEs, to study a class of fully nonlinear integro-differential elliptic equations. Using Krein-Rutmann theorem and Aleksandrov-Bakelman-Pucci estimate, we investigate the existence of principal eigenvalue for a class of fully nonlinear integro-differential elliptic equations. By topological theory and blow-up analysis, we study the existence and multiplicity of solutions for fully nonlinear integro-differential elliptic equations with convex-concave terms. Applying sub- and super-solution method and comparison principle, we study the existence and uniqueness of solutions for nonhomogeneous fully nonlinear integro-differential elliptic equations with Dirichlet boundary condition.

来源于概率论中的偏微分方程（组）不仅具有很强的非线性，而且常是非局部的，它们是偏微分方程研究的前沿热点问题。本项目拟从数学理论研究出发，利用近几年逐步完善的非局部偏微分方程理论来研究一类完全非线性积分-微分椭圆方程。我们利用Krein-Rutmann定理，结合Aleksandrov-Bakelman-Pucci估计，研究完全非线性积分-微分椭圆方程主特征值的存在性。利用拓扑度理论和爆破分析，研究含凹凸非线性项的完全非线性积分-微分椭圆方程解的存在性与多解性。利用上下解方法和比较原理，研究带Dirichlet边界的非齐次完全非线性积分-微分椭圆方程解的存在性和唯一性。

本项目主要研究完全非线性积分-微分（非局部）椭圆方程。项目按计划研究了一类完全非线性积分-微分椭圆方程主特征值的存在性，讨论了含凹凸非线性项的非局部椭圆方程解的存在性与多解性，证明了几类数学物理方程（含有非局部的非线性项）解的存在性和性态。发表相关的学术论文9篇。