The effect of curvature in fractional Hardy--Sobolev inequality with the Spectral Dirichlet Laplacian

TL;DR

This paper establishes the attainability of the optimal constant in a fractional Hardy--Sobolev inequality involving the Spectral Dirichlet Laplacian, under boundary curvature conditions.

Contribution

It proves the existence of extremal functions for the fractional Hardy--Sobolev inequality with boundary singularity, considering boundary curvature effects.

Findings

Best constant in the inequality is attained under boundary concavity.

Boundary curvature influences the attainability of the inequality.

Results extend understanding of fractional inequalities with geometric boundary conditions.

Abstract

We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.