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

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.

Figures (2)

• Figure 1: Image inpainting results of the nonlocal Laplacian equation and the nonlocal biharmonic equation.
• Figure 2: Image inpainting results of the nonlocal $p$-biharmonic equation with different $p$.

Theorems & Definitions (17)

• Definition 2.1
• Theorem 2.2
• proof
• Corollary 2.3
• Lemma 2.4
• Theorem 2.5
• proof
• Lemma 3.1
• Lemma 3.2
• proof