A note on Hardy’s inequalities with boundary singularities

Abstract

Let Ω be a smooth bounded domain in R N with N ≥ 1. In this paper we study the Hardy-Poincaré inequalities with weight function singular at the boundary of Ω. In particular we give sufficient conditions so that the best constant is achieved.

On the Hardy–poincaré Inequality With Boundary Singularities

“…But then Corollary 3.2 implies that v + a = 0 in G + r . Therefore u ≥ R w a for all a ∈ (−1, − 1 2 ) and this contradicts the fact that u |x| ∈ L 2 (Ω) because G + r wa 2 As in [6], starting from exterior domains, we can see that, in general, existence of extremals for µ 0 depends on all the geometry of the domain rather than the geometric constants at the origin. Indeed, let G be a smooth bounded domain of R N , N ≥ 2 with 0 ∈ ∂G.…”

“…For r > 0, set Ω r = B r (0) ∩ (R N \ G). It was shown in [6] that there exits r 1 > 0 such that µ 0 (Ω r ) < N 2 4 for all r ∈ (r 1 , ∞) and µ 0 (Ω r ) is achieved. But Corollary 3.2 and (1.2) yields µ 0 (Ω r ) = N 2 4 for r ∈ (0, r 0 ).…”

“…In the framework of Brezis and Marcus [1], we study the existence and non-existence of minima for the following quotient , in terms of λ ∈ R and Ω. The existence and non-existence of extremals for (1.1) were studied in [3], [4], [6], [7], [8], [12], [13], [14] and the references there in. Especially in [7], the authors proved that for every smooth bounded domain Ω of R N , N ≥ 2, with 0 ∈ ∂Ω where R N + = x ∈ R N : x 1 > 0 , see also Lemma 3.4.…”

Weighted Hardy inequality with higher dimensional singularity on the boundary

“…It amounts to obtain a uniform control of a specific minimizing sequence for µ λ * (Ω, Σ k ) near Σ k via the H 1 -super-solution constructed. We mention that the existence and non-existence of extremals for (1) and related problems were studied in [1,6,7,8,11,12,13,17,18,19] and some references therein. We would like to mention that some of the results in this paper might of interest in the study of semilinear equations with a Hardy potential singular at a submanifold of the boundary.…”

Nonexistence of distributional supersolutions of a semilinear elliptic equation with Hardy potential

“…However it is not valid for every smooth domain. In fact, one can construct a family of smooth bounded domains Ω ε , for which µ(Ω ε ) ≤ (N −2) 2 4 + ε, for ε > 0 small, see [17], [16].…”