图离散p-Laplacian方程及其在图像处理中的应用

The discrete analogue of the p-Laplacian equation on graphs, the so-called discrete p-Laplacian, has been used in various areas including physics, computer science, economics, sociology, etc. Fox example, it can used to model energy flows through a network, vibration of molecules, dynamical systems and also image processing. In this project, we mainly investigate the large time and asymptotic behaviors of the nontrivial solutions of the discrete p-Laplacian with nonlinear terms on graphs. Moreover, we consider the variational problems of the discrete energy on graphs including its sobolev gradient flux. Finally, based on the above theoretical researches, we will design the suitable energy functions on graphs for image processing, such as image segmentation and inpainting and so on.

经典p-Laplacian方程在图上的离散模拟，我们通常称之为图离散 p-Laplacian方程，其已经应用于物理、计算科学、经济以及社会学中的各个领域。例如，模拟网络中的能量流、分子的扰动、动力系统以及图像处理。本项目，我们主要研究带有非线性项的图离散p-Laplacian方程非平凡解的大时间行为以及渐进行为。此外，我们还考虑图离散能量的变分问题，及其相应的Sobolov梯度流。最后，在上述理论研究的基础上，我们将设计合适的图能量泛函，并将其用于图像处理，例如: 图像分割和修补等问题。

本项目主要取得了如下成果：（1）讨论了具有非线性反应项和Dirichlet边界条件的ω-热方程解的爆破现象；（2）讨论了无限的、局部有限的、连通图上带吸收项的ω-热方程的非平凡解的大时间行为；（3）提出了具有反应项的非局部p-Laplacian方程的二值图像分割模型。