# 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} }

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.

## 11 Citations

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## References

SHOWING 1-10 OF 23 REFERENCES

On Hardy inequalities with singularities on the boundary

- Mathematics
- 2011

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

- Mathematics
- 2010

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

- Mathematics
- 2010

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

- MathematicsProceedings 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

- MathematicsProceedings 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…