Nonlocal-to-local convergence of the $p$-Biharmonic evolution equation with the Dirichlet boundary condition

TL;DR

This paper investigates the convergence of nonlocal $p$-biharmonic evolution equations to their local counterparts, establishing existence, uniqueness, and demonstrating applications in image inpainting through numerical experiments.

Contribution

It proves the convergence of nonlocal to local $p$-biharmonic equations with Dirichlet boundary conditions, including existence, uniqueness, and numerical validation.

Findings

Solutions exist and are unique for the nonlocal $p$-biharmonic evolution equation.

Solutions converge to the classical $p$-biharmonic solutions as the kernel is rescaled.

Numerical experiments confirm the effectiveness of the nonlocal model in image inpainting.

Abstract

This paper studies the nonlocal $p$-biharmonic evolution equation with the Dirichlet boundary condition that arises in image processing and data analysis. We prove the existence and uniqueness of solutions to the nonlocal equation and discuss the large time behavior of the solution. By appropriately rescaling the nonlocal kernel, we further show that the solution converges to the solution of the classical $p$-biharmonic equation with the Dirichlet boundary condition. Numerical experiments are presented to demonstrate the effectiveness of the nonlocal $p$-biharmonic equation for image inpainting.