Dimension reduction for $-\Delta_1$

TL;DR

This paper develops a 3D-2D dimension reduction for the $- abla_1$ operator and explores the asymptotic behavior of $- abla_p$ as $p$ approaches 1, using $\Gamma$-convergence, duality, and least gradient functions.

Contribution

It introduces a novel 3D-2D reduction for $- abla_1$ and provides a power law approximation for $- abla_p$ as $p$ approaches 1, advancing understanding of these operators.

Findings

Established a 3D-2D dimension reduction for $- abla_1$.

Derived a power law approximation for $- abla_p$ as $p o 1$.

Utilized $\Gamma$-convergence, duality, and asymptotics for least gradient functions.

Abstract

A 3D-2D dimension reduction for $-\Delta_1$ is obtained. A power law approximation from $-\Delta_p$ as $p \to 1$ in terms of $\Gamma$- convergence, duality and asymptotics for least gradient functions has also been provided.