Remarks on extremals of sharp Sobolev trace inequalities on the unit balls

TL;DR

This paper explicitly characterizes extremals of certain sharp Sobolev trace inequalities on unit balls and classifies solutions to a conformally covariant bi-harmonic boundary value problem in four dimensions.

Contribution

It provides explicit extremal functions for fourth-order sharp trace inequalities and classifies solutions to a related bi-harmonic boundary problem.

Findings

Explicit forms for extremals of sharp trace inequalities

Classification of bi-harmonic equation solutions with conformal boundary conditions

Advances understanding of conformally covariant boundary problems

Abstract

We show explicit forms for extremals of some fourth-order sharp trace inequalities on the unit balls recently proved by Ache-Chang. We also give a classification result of the bi-harmonic equation on $\mathbb{R}^4_+$ with some conformally covariant boundary conditions.