The role of absorption terms in Dirichlet problems for the prescribed mean curvature equation

TL;DR

This paper investigates how absorption terms influence the existence, regularity, and uniqueness of solutions to Dirichlet problems involving prescribed mean curvature equations, especially for low-regularity data.

Contribution

It introduces new conditions under which solutions exist for minimal regularity data and establishes sharp bounds and uniqueness results for these solutions.

Findings

Existence of solutions for data in L^1(Ω) without smallness constraints.

Sharp boundedness results for data in L^N(Ω).

Uniqueness of solutions when g is increasing.

Abstract

In this paper we study existence and uniqueness of solutions to Dirichlet problems as $$ \begin{cases} g(u) -{\rm div}\left(\frac{D u}{\sqrt{1+|D u|^2}}\right) = f & \text{in}\;\Omega,\\ \newline u=0 & \text{on}\;\partial\Omega, \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$ ($N\geq 2$) with Lipschitz boundary, $g:\mathbb{R}\to\mathbb{R}$ is a continuous function and $f$ belongs to some Lebesgue spaces. In particular, under suitable saturation and sign assumptions, we explore the regularizing effect given by the absorption term $g(u)$ in order to get a solutions for data $f$ merely belonging to $L^1(\Omega)$ and with no smallness assumptions on the norm. We also prove a sharp boundedness result for data in $L^{N}(\Omega)$ as well as uniqueness if $g$ is increasing.