A Class Of Elliptic Equations with Interior Degeneration

TL;DR

This paper investigates a class of linear degenerate elliptic equations with interior degeneration, highlighting the challenges in their interpretation and numerical resolution, which may also impact nonlinear variants.

Contribution

It introduces a class of elliptic equations with interior degeneration and discusses the interpretative and numerical challenges associated with them.

Findings

No unique interpretation of the equations is possible.

Numerical schemes exhibit complex behavior due to degeneration.

Potential issues extend to nonlinear versions.

Abstract

A class of linear degenerate elliptic equations inspired by nonlinear diffusions of image processing is considered. It is characterized by an interior degeneration of the diffusion coefficient. It is shown that no particularly natural, unique interpretation of the equation is possible. This phenomenon is reflected in the behavior of numerical schemes for its resolution and points to similar issues that potentially affect its nonlinear counterpart.