# Weighted Hardy inequality with higher dimensional singularity on the boundary

@article{Fall2012WeightedHI,
title={Weighted Hardy inequality with higher dimensional singularity on the boundary},
author={Mouhamed Moustapha Fall and Fethi Mahmoudi},
journal={Calculus of Variations and Partial Differential Equations},
year={2012},
volume={50},
pages={779-798}
}
• Published 31 January 2012
• Mathematics
• Calculus of Variations and Partial Differential Equations
Let $$\Omega$$Ω be a smooth bounded domain in $$\mathbb R ^N$$RN with $$N\ge 3$$N≥3 and let $$\Sigma _k$$Σk be a closed smooth submanifold of $$\partial \Omega$$∂Ω of dimension $$1\le k\le N-2$$1≤k≤N-2. In this paper we study the weighted Hardy inequality with weight function singular on $$\Sigma _k$$Σk. In particular we provide necessary and sufficient conditions for existence of minimizers.
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## References

SHOWING 1-10 OF 23 REFERENCES
On Hardy inequalities with singularities on the boundary
Abstract In this Note we present some Hardy–Poincare inequalities with one singularity localized on the boundary of a smooth domain. Then, we consider conical domains in dimension N ⩾ 3 whose vertex
ON THE HARDY–POINCARÉ INEQUALITY WITH BOUNDARY SINGULARITIES
Let Ω be a smooth bounded domain in ℝN with N ≥ 1. In this paper we study the Hardy–Poincare inequality with weight function singular at the boundary of Ω. In particular we provide sufficient and
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–Poincare inequalities with weight function singular at the boundary of Ω . In particular we give
Hardy—Poincaré inequalities with boundary singularities
• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2012
We are interested in variational problems involving weights that are singular at a point of the boundary of the domain. More precisely, we study a linear variational problem related to the Poincaré
Existence of minimizers for Schrödinger operators under domain perturbations with application to Hardy's inequality
• Mathematics
• 2004
The paper studies the existence of minimizers for Rayleigh quotients formula math. where Q is a domain in R N , and V is a nonzero nonnegative function that may have singularities on ∂Ω. As a model
Large Solutions to Semilinear Elliptic Equations with Hardy Potential and Exponential Nonlinearity
• Mathematics
• 2010
On a bounded smooth domain Ω ⊂ ℝ N , we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to ∂ ⊂. We derive global a
Critical Hardy–Sobolev inequalities
• Mathematics
• 2006
Abstract We consider Hardy inequalities in R n , n ⩾ 3 , with best constant that involve either distance to the boundary or distance to a surface of co-dimension k n , and we show that they can still
On the structure of Hardy–Sobolev–Maz'ya inequalities
• Mathematics
• 2008
We establish new improvements of the optimal Hardy inequality in the half-space. We first add all possible linear combinations of Hardy type terms, thus revealing the structure of this type of
Concentration on minimal submanifolds for a singularly perturbed Neumann problem
• Physics, Mathematics
• 2006
Abstract We consider the equation − e 2 Δ u + u = u p in Ω ⊆ R N , where Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N ⩾
On a class of two-dimensional singular elliptic problems
• Mathematics
Proceedings of the Royal Society of Edinburgh: Section A Mathematics
• 2001
We consider Dirichlet problems of the form −|x|αΔu = λu + g(u) in Ω, u = 0 on ∂Ω, where α, λ ∈ R, g ∈ C(R) is a superlinear and subcritical function, and Ω is a domain in R2. We study the existence