Sobolev inequalities for Neumann Laplacians on half spaces

TL;DR

This paper investigates Sobolev inequalities for various fractional Neumann Laplacians on half-spaces, focusing on the attainability of Sobolev constants for these operators.

Contribution

It introduces and analyzes different fractional Neumann Laplacians on half-spaces and studies the conditions under which Sobolev constants are attained.

Findings

Attainability of Sobolev constants varies among the different fractional Neumann Laplacians.

Provides new insights into Sobolev inequalities for fractional operators on half-spaces.

Establishes conditions for the existence of extremal functions for these inequalities.

Abstract

We consider different fractional Neumann Laplacians of order s, 0<s<1, namely, the Restricted Neumann Laplacian, the Semirestricted Neumann Laplacian and the Spectral Neumann Laplacian. In particular, we are interested in attainability of Sobolev constants for these operators in half-spaces.