The role of the mean curvature in a Hardy–Sobolev trace inequality

Abstract

The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.

Borderline Variational Problems Involving Fractional Laplacians and Critical Singularities

“…The non-local case has also been the subject of several studies, but in the absence of the Hardy term, i.e., when γ = 0. In [16], Fall, Minlend and Thiam proved the existence of extremals for µ 0,s (R n ) in the case α = 1. Recently, J. Yang in [37] proved that there exists a positive, radially symmetric and non-increasing extremal for µ 0,s (R n ) when α ∈ (0, 2).…”

“…|x| s (see [10], [16], [37] and the references therein). These cases were also studied on smooth bounded domains (see for example [2], [3], [4], [15], [34] and the references therein).…”

Hardy-Sobolev inequality with singularity a curve

“…The first paper, to our knowledge, being the one of Ghoussoub and Kang [17] who considered the Hardy-Sobolev inequality with singularity at the boundary. For more results in this direction, see the works of Ghoussoub and Robert in [12][13][14]16], Demyanov and Nazarov [8], Chern and Lin [7], Lin and Li [19], the authors and Minlend in [11] and the references there in. We point out that in the pure Hardy-Sobolev case, σ ∈ (0, 2), with singularity at the boundary, one has existence of minimizers for every dimension N ≥ 3 as long as the mean curvature of the boundary is negative at the point singularity, see [15].…”