On fractional Laplacians -- 3

TL;DR

This paper explores the influence of dilation groups on fractional Laplacians and applies findings to analyze the Brezis--Nirenberg effect in boundary value problems involving the Navier-Laplacian.

Contribution

It provides new insights into the role of dilation groups on fractional Laplacians and applies these to study boundary value problems with the Navier-Laplacian.

Findings

Dilation groups significantly affect quadratic forms of fractional Laplacians.

Results elucidate the Brezis--Nirenberg effect in specific boundary value problems.

New relationships between fractional Dirichlet and Navier Laplacians are established.

Abstract

We investigate the role of the noncompact group of dilations in $\mathbb R^n$ on the difference of the quadratic forms associated to the fractional Dirichlet and Navier Laplacians. Then we apply our results to study the Brezis--Nirenberg effect in two families of noncompact boundary value problems involving the Navier-Laplacian.