库仑系统中奇异性方程的定性分析

This proposal is concerned about several PDEs involving singular terms in the Coulomb system, including the singular equations with the Coulomb potentials, the discrete Hardy-Littlewood-Sobolev-type equations, and the Coulomb-Sobolev-type equations. Those equations are focuses of attention now because they are widely applied in quantum mechanics, N-body dynamics, computational chemistry and other fields. In addition, the study of such a class of equations also promotes the progress of other mathematical branches such as the nonlinear analysis, the classical inequalities and the geometric analysis. We are going to study the qualitative properties such as existence, integrability, regularity, sigularity estimates and decay estimates, and classification of positive solutions. Such a qualitative analysis can help us to describe the shapes of solutions and well understand the statistic sense of wave functions and density functionals.

本课题研究库仑系统中几类奇异偏微分方程，包括含有库仑位势的奇异方程，离散的Hardy-Littlewood-Sobolev型方程和Coulomb-Sobolev型方程。这些方程广泛出现在量子力学，多体动力学，以及计算化学等学科中，如今已受到越来越多的关注。在数学领域内，此类方程的研究也促进了非线性分析，经典不等式和几何分析的发展。我们将讨论方程解的定性性质，包括存在性，可积性，正则性，奇异性估计和衰减估计，以及正解的分类等。这些定性性质能帮助我们刻画解的形状，并更深入地了解波函数和密度泛函的统计学本质。

本课题围绕库仑系统中两类问题展开研究。一是研究含有库仑位势的微分方程（包括Choquard方程，Hartree方程和Schrodinger-Poisson方程）。另一个是研究对估计库仑能量起重要作用的不等式（包括Hardy-Littlewood-Sobolev不等式以及Coulomb-Sobolev不等式）的极值函数满足的Euler-Lagrange方程，它们分别对应着含有Riesz位势的Lane-Emden型积分方程和Schrodinger-Poisson-Slater方程。我们应用已有的方法处理上述非局部问题外，也建立和发展了一些新的技巧。我们在方程解的存在性，正则性，无穷远处的极限行为等方面取得一系列成果。此外，我们还对含有Riesz位势的相变方程做了初步探索。