Submitted Paper

Inserted: 23 jul 2026
Last Updated: 18 aug 2026

Year: 2026

Abstract:

In this paper, we study the Hölder regularity of the solution $u$ to the following least gradient problem with nonhomogeneous term \[\begin{equation} \label{LGP} \inf \bigg\{ \int_{\Omega} |Du| + \lambda\int_{\Omega} u\,\,: \,\, u \in BV(\Omega),\,\,\,\, u = f\,\,\,\mbox{on}\,\,\,\partial\Omega \bigg\}, \end{equation}\] in terms of the regularity of the boundary datum $f$. Inspired by $[5,9]$, we show that under some assumptions on the mean curvature of $\Omega$ and the constant $\lambda$, we have the following statements: \[\begin{equation} \label{statements} \begin{cases} f \in C^{0,\alpha}(\partial\Omega) \Rightarrow u \in C^{0,\frac{\alpha}{2}}(\overline{\Omega}),\\ f \in C^{1,\alpha}(\partial\Omega) \Rightarrow u \in C^{0,\frac{1+\alpha}{2}}(\overline{\Omega}). \end{cases} \end{equation}\]

Keywords: Least gradient problem, Nonhomogeneous term, 1−Laplacian

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