On the Sobolev and Hardy constants for the fractional Navier Laplacian

TL;DR

This paper establishes that the Sobolev and Hardy constants are identical for both Dirichlet and Navier fractional Laplacians of any order between 0 and n/2 on bounded domains, unifying their spectral properties.

Contribution

It proves the equivalence of Sobolev and Hardy constants for Dirichlet and Navier fractional Laplacians across all relevant orders, extending previous results to a broader setting.

Findings

Sobolev and Hardy constants coincide for Dirichlet and Navier fractional Laplacians

The result holds for any real order m in (0, n/2)

Unifies spectral properties of fractional Laplacians on bounded domains

Abstract

We prove the coincidence of the Sobolev and Hardy constants relative to the "Dirichlet" and "Navier" fractional Laplacians of any real order $m\in(0,\frac{n}{2})$ over bounded domains in $\mathbb R^n$.