A simple proof of attainability for the Sobolev inequality

TL;DR

This paper presents a straightforward proof demonstrating the existence of a minimizer for the Sobolev inequality, utilizing a representation formula based on a cut-off fundamental solution.

Contribution

It introduces a simplified method for proving the attainability of the Sobolev inequality's minimizer using a novel representation approach.

Findings

Established the existence of minimizers for the Sobolev inequality.

Provided a new proof technique based on a cut-off fundamental solution.

Simplified the theoretical understanding of Sobolev inequality attainability.

Abstract

We give a simple proof of the existence of a minimizer for the Sobolev inequality. Our proof is based on a representation formula via a cut-off fundamental solution.