An overview on extremals and critical points of the Sobolev inequality in convex cones

TL;DR

This survey explores the sharp Sobolev inequality in convex cones, employing optimal transport methods, analyzing the associated Euler-Lagrange equations, and discussing stability results to deepen understanding of extremals and critical points.

Contribution

It provides a comprehensive overview of the Sobolev inequality in convex cones, including new proofs via optimal transport and insights into stability and critical point analysis.

Findings

Proof of the sharp Sobolev inequality using optimal transport

Analysis of the Euler-Lagrange equation related to the inequality

Discussion of stability results for extremals

Abstract

In this survey, we consider the sharp Sobolev inequality in convex cones. We also prove it by using the optimal transport technique. Then we present some results related to the Euler-Lagrange equation of the Sobolev inequality: the so-called critical p-Laplace equation. Finally, we discuss some stability results related to the Sobolev inequality.