A general quasilinear elliptic problem with variable exponents and Neumann boundary conditions for image processing

TL;DR

This paper establishes existence and uniqueness results for a broad class of variable exponent elliptic problems with Neumann boundary conditions, using innovative techniques to overcome coercivity issues, with applications in image denoising.

Contribution

It introduces a novel method to prove existence and uniqueness for non-coercive elliptic problems with variable exponents, applicable to multiple-phase scenarios.

Findings

Proved existence and uniqueness of solutions under weak assumptions.

Extended results to multiple-phase elliptic problems.

Developed an innovative approach overcoming lack of coercivity.

Abstract

The aim of this paper is to state and prove existence and uniqueness results for a general elliptic problem with homogeneous Neumann boundary conditions, often associated with image processing tasks like denoising. The novelty is that we surpass the lack of coercivity of the Euler-Lagrange functional with an innovative technique that has at its core the idea of showing that the minimum of the energy functional over a subset of the space $W^{1,p(x)}(\Omega)$ coincides with the global minimum. The obtained existence result applies to multiple-phase elliptic problems under remarkably weak assumptions.