Weighted Hardy inequality on Riemannian manifolds

Abstract

Let [Formula: see text] be a smooth compact Riemannian manifold of dimension [Formula: see text] and let [Formula: see text] to be a closed submanifold of dimension [Formula: see text]. In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on [Formula: see text] within the framework of Brezis–Marcus–Shafrir [Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000) 177–191]. In particular, we provide necessary and sufficient conditions for existence of minimizers.

Hardy–Sobolev equations on compact Riemannian manifolds

“…In this case, up to taking α close enough to α 1 , we get that f u ∈ L p (M ), for all p < n N ′ s . Therefore, (23), N ′ s ∈ (1, 2) (because N ′ s > 0 and s < 2), and standard elliptic theory yield u ∈ C 0,θ (M ) for all θ < 2 − N ′ s . It then follows from the definition of α 1 that α 1 ≥ 2 − N ′ s .…”

“…and α ∈ (0, 1). The existence of a minimizer of J in H 2 1 (M ) \ {0} has been proved independently by Thiam [23].…”

Sharp Hardy inequalities via Riemannian submanifolds

“…As far as we know, Carron [11] was the first who studied weighted L 2 -Hardy inequalities on Riemannian manifolds. Inspired by [11], a systematic study is carried out by Berchio, Ganguly and Grillo [5], D'Ambrosio and Dipierro [17], Kombe and Özaydin [33,34], Thiam [42], Yang, Su and Kong [46], etc. In particular, (1.1) has been successfully generalized to complete non-compact Riemannian manifolds with non-positive sectional curvature.…”

Hardy-Sobolev inequality with singularity a curve

“…In this case, for N ≥ 4 it is enough that h(y 0 ) < 0 to get a minimizer, whereas for N = 3, the problem is no more local and existence of minimizers is guaranteed by the positiveness of a certain mass -the trace of the regular part of the Green function of the operator −∆ + h with zero Dirichlet data, see Druet [9]. For σ = 2, the problem reduces to a linear eigenvalue problem with Hardy potential, existence and nonexistence results were obtained by the second author in [25]. Here, we deal with the case σ ∈ (0, 2).…”