On fractional Laplacians -- 2

TL;DR

This paper compares two types of fractional Laplacians, Navier and Dirichlet, for s > -1, and provides variational characterizations for s in (-1,0), advancing the theoretical understanding of these operators.

Contribution

It introduces dual variational characterizations of Navier and Dirichlet fractional Laplacians for negative s, extending previous work and clarifying their relationship.

Findings

Comparison of Navier and Dirichlet fractional Laplacians for s > -1.

Dual variational characterizations for s in (-1,0).

Enhanced understanding of fractional Laplacian operators.

Abstract

The present paper is the natural evolution of arXiv:1308.3606. For $s>-1$ we compare two natural types of fractional Laplacians $(-\Delta)^s$, namely, the "Navier" and the "Dirichlet" ones. As a main tool, we give the "dual" Caffarelli--Silvestre and Stinga--Torrea variational characterizations of these operators for $s\in(-1,0)$.