Optimal global $BV$ regularity for 1-Laplace type BVP's with singular lower order terms

TL;DR

This paper characterizes the regularity of solutions to a 1-Laplacian boundary value problem with singular lower order terms, establishing existence and regularity conditions depending on the behavior of the singular function and data integrability.

Contribution

It extends the regularity results for 1-Laplacian problems with singular terms, generalizing the Lazer-McKenna result to the 1-Laplacian case without growth restrictions at zero.

Findings

Existence of finite energy BV solutions under sharp conditions.

Regularity depends on the behavior of the singular function at infinity.

Behavior at zero does not affect regularity in contrast to p-Laplacian cases.

Abstract

In this paper we provide a complete characterization of the regularity properties of the solutions associated to the homogeneous Dirichlet problem \begin{equation*} \begin{cases} \displaystyle - \Delta_1 u= h(u)f & \text{in } \Omega, \\ \newline u=0 & \text{on } \partial \Omega, \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^N$ is a bounded open set with Lipschitz boundary, $f \in L^m(\Omega)$ with $m\geq 1$ is a nonnegative function and $h\colon \mathbb{R}^+ \to \mathbb{R}^+$ is continuous, possibly singular at the origin and bounded at infinity. Without any growth restrictions on $h$ at zero, we prove existence of global finite energy solutions in $BV(\Omega)$ under sharp conditions on the summability of $f$ and on the behaviour of $h$ at infinity. Roughly speaking, the faster $h$ goes to zero at infinity, the less regularity is required on $f$. In contrast to the $p$-Laplacian case ($p>1$), we show that the behaviour of $h$ at the origin plays essentially no role. The main result contains an extension of the celebrated one of Lazer-McKenna (\cite{lm}) to the case of the $1$-Laplacian as principal operator.