A note on the Gagliardo-Nirenberg inequality in a bounded domain

TL;DR

This paper provides a simplified proof of the classical Gagliardo-Nirenberg inequality in bounded domains and introduces a new inequality for Lipschitz domains, expanding the theoretical understanding of these inequalities.

Contribution

It offers a straightforward proof of the existing inequality and establishes a new Gagliardo-Nirenberg inequality specifically for bounded Lipschitz domains.

Findings

Simplified proof of Gagliardo-Nirenberg inequality in bounded domains

Introduction of a new inequality for Lipschitz domains

Enhanced theoretical framework for inequalities in bounded regions

Abstract

The classical Gagliardo-Nirenberg inequality was established in $\mathbb{R}^n$. An extension to a bounded domain was given by Gagliardo in 1959. In this note, we present a simple proof of this result and prove a new Gagliardo-Nirenberg inequality in a bounded Lipschitz domain.