Well-posedness of a nonlinear integro-differential problem and its rearranged formulation

TL;DR

This paper investigates the existence and uniqueness of solutions for a nonlinear integro-differential problem, introduces a rearranged formulation for dimensional reduction, and demonstrates its effectiveness in image processing tasks through numerical methods.

Contribution

It presents a novel rearranged formulation of the nonlinear problem, providing theoretical insights and a fast numerical method for practical image processing applications.

Findings

Successful dimensional reduction of the problem

Proven existence and uniqueness of solutions

Effective numerical implementation for image filtering and segmentation

Abstract

We study the existence and uniqueness of solutions of a nonlinear integro-differential problem which we reformulate introducing the notion of the decreasing rearrangement of the solution. A dimensional reduction of the problem is obtained and a detailed analysis of the properties of the solutions of the model is provided. Finally, a fast numerical method is devised and implemented to show the performance of the model when typical image processing tasks such as filtering and segmentation are performed.