Elliptic equations involving the 1--Laplacian and a total variation term with $L^{N,\infty}$--data

TL;DR

This paper investigates a nonlinear elliptic equation involving the 1-Laplacian and total variation with data in a Marcinkiewicz space, exploring existence, uniqueness, and regularity of unbounded solutions and effects of different gradient terms.

Contribution

It introduces a suitable notion of unbounded solutions for the 1-Laplacian with Marcinkiewicz data and analyzes how different gradient terms influence solution properties.

Findings

Existence of unbounded solutions with $L^{N, abla}$ data.

Regularizing effects depend on the gradient term $g$.

Examples of explicit solutions are provided.

Abstract

In this paper we study, in an open bounded set $\Omega\subset\mathbb R^N$ with Lipschitz boundary $\partial\Omega$, the Dirichlet problem for a nonlinear singular elliptic equation involving the $1$--Laplacian and a total variation term, that is, the inhomogeneous case of the equation appearing in the level set formulation of the inverse mean curvature flow. Our aim is twofold. On the one hand, we consider data belonging to the Marcinkiewicz space $L^{N,\infty}(\Omega)$, which leads to unbounded solutions. So, we have to begin introducing the suitable notion of unbounded solution to this problem. Moreover, examples of explicit solutions are shown. On the other hand, this equation allows us to deal with many related problems having a different gradient term. It is known that the total variation term induces a regularizing effect on existence, uniqueness and regularity. We focus on analyzing whether those features remain true when general gradient terms are taken. Roughly speaking, the bigger $g$, the better the properties of the solution.