A minimization procedure to the existence of segregated solutions to parabolic reaction-diffusion systems

Competition in periodic media: III -- Existence \& stability of segregated periodic coexistence states

In this paper we consider a system of parabolic reaction-diffusion equations with strong competition and two related scalar reaction-diffusion equations. We are mainly concerned with the case of periodic coefficients and periodic solutions. We show that, for sufficiently large periods, these models have stationary, non-constant, fully non-trivial and stable solutions. We compare our results with already known results about the existence and non-existence of such solutions. Finally, we provide ecological interpretations for these results in terms of resistance against an invasion.

部分分离椭圆系统的 Gamma 收敛

Gamma Convergence of Partially Segregated Elliptic Systems

本文利用惩罚能量泛函研究部分分离椭圆系统，证明平面情形下惩罚能量 Gamma 收敛于严格分离的受约束 Dirichlet 能量。

高维情况下临界椭圆系统的分离解

Segregated Solutions to Critical Elliptic Systems in High Dimensions ($N \geq 5$)

本文研究高维临界椭圆系统中多重分离解的存在性，通过构造完备度量空间和尾最小化论证，证明了在特定条件下存在无限多个非径向分离解。

具有漂移的退化交叉扩散系统的分离解

Segregated solutions of a degenerate cross-diffusion system with drifts

通过拉格朗日公式和极小移动方案，证明了一维带独立漂移的退化交叉扩散系统全局弱解的存在性，并处理了多孔介质退化、对数熵和快速扩散情形。

多相四次域（存在性和唯一性）

Multiphase quadrature domains (existence and uniqueness)

本文研究多相四次域的存在性和唯一性，通过变分框架证明子调和函数的四次域存在性，并探讨能量泛函的最小化与约束条件的关系。

基于各向异性单色图像处理与方向自动调整的梯度系统

A gradient system based on anisotropic monochrome image processing with orientation auto-adjustment

本文研究各向异性单色图像处理中带方向自动调整的伪抛物线偏微分方程组，通过降低导数阶数提升计算效率并建立稳定极小化过程，证明了系统的适定性及图像去噪的稳定性。