Extremal functions for an embedding from some anisotropic space, and partial differential equation involving the "one Laplacian"

TL;DR

This paper establishes the existence of extremal functions for anisotropic space embeddings with some Sobolev exponents equal to 1 and shows these extremals satisfy a PDE involving the 1-Laplacian.

Contribution

It introduces new extremal functions for anisotropic space embeddings and links them to a PDE with the 1-Laplacian, expanding understanding of such embeddings.

Findings

Existence of extremal functions proven

Extremal functions satisfy a PDE with the 1-Laplacian

Extension to cases with Sobolev exponents equal to 1

Abstract

In this paper, we prove the existence of extremal functions for the best constant of embedding from anisotropic space, allowing some of the Sobolev exponents to be equal to $1$. We prove also that the extremal functions satisfy a partial differential equation involving the $1$ Laplacian.