Preprint
Inserted: 9 dec 2025
Last Updated: 4 feb 2026
Year: 2025
Abstract:
Given a domain $\Omega \subset \mathbb{R}^N$, we consider the following generalization of the least gradient problem:
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\inf \bigg\{ \int_{\Omega} |Du| +\int_{\Omega} \psi\,u\,\,: \,\, u \in BV(\Omega),\,\,\,\, u = f\,\,\,\mbox{on}\,\,\,\partial\Omega \bigg\},$
where $\psi \in L^N(\Omega)$ and $f \in L^1(\partial\Omega)$ are given functions. If the domain $\Omega$ satisfies a $\psi-$barrier condition and the boundary datum $f$ is continuous, we show existence of a solution $u$ to this problem provided that $\psi \in L^p(\Omega)$ with $p>N$ and $\psi$ is sufficiently small in an appropriate sense.
By a comparison theorem, we will also prove uniqueness of the solution provided that $\psi$ is a constant $\lambda$ and $|\lambda|$ is smaller than the Cheeger constant of $\Omega$. When $N \leq 7$, we show that the solution $u$ is continuous on $\overline{\Omega}$. These generalize previous results obtained for the classical BV least gradient problem (without the nonhomogeneous term $\psi$) in $
[$17,9,8$
]$.
Keywords: 1-laplacian, least gradient problem, Nonhomogeneous term
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