A note on Hardy's inequalities with boundary singularities

2016

Let Ω be a bounded domain in R N with 0 ∈ ∂Ω and N ≥ 2. In this paper we study the Hardy-Poincaré inequality for maps in H 1 0 (Ω). In particular we give sufficient and some necessary conditions so that the best constant is achieved.

Journal de Mathématiques Pures et Appliquées, 2018

We study the Hardy inequality when the singularity is placed on the boundary of a bounded domain in R n that satisfies both an interior and exterior ball condition at the singularity. We obtain the sharp Hardy constant n 2 /4 in case the exterior ball is large enough and show the necessity of the large exterior ball condition. We improve Hardy inequality with the best constant by adding a sharp Sobolev term. We next produce criteria that lead to characterizing maximal potentials that improve Hardy inequality. Breaking the criteria one produces successive improvements with sharp constants. Our approach goes through in less regular domains, like cones. In the case of a cone, contrary to the smooth case, the Sobolev constant does depend on the opening of the cone.

2003

In this note, we present some Hardy type inequalities for functions which do not vanish on the boundary of a given domain. We establish these inequalities for both bounded and unbounded domains and also obtain the best embedding constants in these inequalities for special domains. Our results are motivated by and building upon some recent work in [5, 6, 9, 12]. 2000 Mathematics Subject Classification. 35J20, 35J25. Key words and phrases. Hardy inequality with boundary terms, weighted versions of Hardy inequality.

arXiv (Cornell University), 2022

This is the first part of our research on certain sharp Hardy-Sobolev inequalities and the related elliptic equations. In this part we shall establish some sharp weighted Hardy-Sobolev inequalities whose weights are distance functions to the boundary.

Comptes Rendus Mathematique, 2004

Let Ω be a smooth bounded domain in IR N , N ≥ 3. We show that Hardy's inequality involving the distance to the boundary, with best constant (1/4), may still be improved by adding a multiple of the critical Sobolev norm.

Transactions of the American Mathematical Society, Series B, 2022

There have been numerous studies on Hardy’s inequality on a bounded domain, which holds for functions vanishing on the boundary. On the other hand, the classical Legendre differential equation defined in an interval can be regarded as a Neumann version of the Hardy inequality with subcritical weight functions. In this paper we study a Neumann version of the Hardy inequality on a bounded C 2 C^2 -domain in R n \mathbb {R}^n of the following form ∫ Ω d β ( x ) | ∇ u ( x ) | 2 d x ≥ C ( α , β ) ∫ Ω | u ( x ) | 2 d α ( x ) d x with ∫ Ω u ( x ) d α ( x ) d x = 0 , \begin{equation*} \int _\Omega d^{\beta }(x) |\nabla u(x) |^2 dx \ge C(\alpha ,\beta ) \int _\Omega \frac {|u(x)|^2}{d^{\alpha }(x)} dx \quad \text { with }\quad \int _\Omega \frac {u(x)}{d^{\alpha }(x)} dx=0, \end{equation*} where d ( x ) d(x) is the distance from x ∈ Ω x \in \Omega to the boundary ∂ Ω \partial \Omega and α , β ∈ R \alpha ,\beta \in \mathbb {R} . We classify all ( α , β ) ∈ R 2 (\alpha ,\beta ) \in \mathbb {...

International Mathematical Series, 2009

In this article we first establish a complete characterization of Hardy's inequalities in R n involving distances to different codimension subspaces. In particular the corresponding potentials have strong interior singularities. We then provide necessary and sufficient conditions for the validity of Hardy-Sobolev-Maz'ya inequalities with optimal Sobolev terms.

Transactions of the American Mathematical Society, 2003

We show that the classical Hardy inequalities with optimal constants in the Sobolev spaces W 1,p 0 and in higher-order Sobolev spaces on a bounded domain Ω ⊂ R n can be refined by adding remainder terms which involve L p norms. In the higher-order case further L p norms with lower-order singular weights arise. The case 1 < p < 2 being more involved requires a different technique and is developed only in the space W 1,p 0 .

Journal of Inequalities and Applications, 2012

We investigate the k-th order Hardy inequality (1.1) for functions satisfying rather general boundary conditions (1.2), show which of these conditions are admissible and derive sufficient, and necessary and sufficient, conditions (for 0 &lt; q &lt; ∞, p &gt; 1) on u, v for (1.1) to hold.

Communications in Contemporary Mathematics, 2015

Let Ω ⊂ ℝN be a bounded regular domain of ℝN and 1 &lt; p &lt; ∞. The paper is divided into two main parts. In the first part, we prove the following improved Hardy inequality for convex domains. Namely, for all [Formula: see text], we have [Formula: see text] where d(x) = dist (x, ∂Ω), [Formula: see text] and C is a positive constant depending only on p, N and Ω. The optimality of the exponent of the logarithmic term is also proved. In the second part, we consider the following class of elliptic problem [Formula: see text] where 0 &lt; q ≤ 2* - 1. We investigate the question of existence and nonexistence of positive solutions depending on the range of the exponent q.