The Dirichlet Problem For the Logarithmic p-Laplacian

TL;DR

This paper introduces the logarithmic p-Laplacian operator, explores its spectral properties, establishes inequalities, and discusses maximum principles, providing a new framework for nonlocal nonlinear PDEs with logarithmic order.

Contribution

It defines the logarithmic p-Laplacian, develops a variational framework, and connects its spectral properties to those of the fractional p-Laplacian, including inequalities and principles.

Findings

Established a variational framework for the logarithmic p-Laplacian.

Proved a Faber-Krahn inequality for the first eigenvalue.

Demonstrated boundary Hardy-type inequalities and eigenfunction boundedness.

Abstract

We introduce and study the logarithmic $p$-Laplacian $L_{\Delta_p}$, which emerges from the formal derivative of the fractional $p$-Laplacian $(-\Delta_p)^s$ at $s=0$. This operator is nonlocal, has logarithmic order, and is the nonlinear version of the newly developed logarithmic Laplacian operator. We present a variational framework to study the Dirichlet problems involving the $L_{\Delta_p}$ in bounded domains. This allows us to investigate the connection between the first Dirichlet eigenvalue and eigenfunction of the fractional $p$-Laplacian and the logarithmic $p$-Laplacian. As a consequence, we deduce a Faber-Krahn inequality for the first Dirichlet eigenvalue of $L_{\Delta_p}$. We discuss maximum and comparison principles for $L_{\Delta_p}$ in bounded domains and demonstrate that the validity of these depends on the sign of the first Dirichlet eigenvalue of $L_{\Delta_p}$. In addition, we prove that the first Dirichlet eigenfunction of $L_{\Delta_p}$ is bounded. Furthermore, we establish a boundary Hardy-type inequality for the spaces associated with the weak formulation of the logarithmic $p$-Laplacian.