Sobolev Trace Inequalities

TL;DR

This paper investigates extremal functions for Sobolev trace inequalities, proving their existence under symmetry conditions and computing the best constant in a limiting case, advancing understanding of these inequalities.

Contribution

It establishes the existence of extremal functions for Sobolev trace inequalities and computes the optimal constant in a specific limiting case.

Findings

Existence of extremal functions proved using concentration compactness.

The conjectured extremal function with conformal symmetry is validated.

The best constant for a limiting case of the inequalities is explicitly calculated.

Abstract

The existence of extremal functions for the Sobolev trace inequalities is studied using the concentration compactness theorem. The conjectured extremal, the function of conformal factor, is considered and is proved to be an actual extremal function with extra symmetry condition on functions. One of the limiting cases of the Sobolev trace inequalities is investigated and the best constant for this case is computed.