A unified approach to improved L p Hardy inequalities with best constants

By G. Barbatis, Stathis Filippas, A. Tertikas. 2003

297 citations recorded by the source corpus.

We present a unified approach to improved L p Hardy inequalities in R N . We consider Hardy potentials that involve either the distance from a point, or the distance from the boundary, or even the intermediate case where the distance is taken from a surface of codimension 1 < k < N. In our main result, we add to the right hand side of the classical Hardy inequality a weighted L p norm with optimal weight and best constant. We also prove nonhomogeneous improved Hardy inequalities, where the right hand side involves weighted L q norms, q ¬= p.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/51d6e527136276fedfc2b471768e4591e8e66aae

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Citations in the corpus

Improved Hardy and Rellich inequalities on Riemannian manifolds (2007, background): “A detailed treatment of improved Hardy inequalities with best constants, involving various kinds of distance functions in the Euclidean space R n , can be found in [4].”
A geometric characterization of a sharp Hardy inequality (2011, unknown): “In the case p = 2, Brezis and Marcus [7] proved the following inequality for bounded domains with C2 boundary (1.8) ∫ Ω |∇u|2dx ≥ 1 4 ∫ Ω (u/δ)2dx+ Λ ∫ Ω u2dx,∀u ∈ H10 (Ω), where Λ is the best constant defined as (1.9) Λ := inf∫ Ω f 2dx=1 [ ∫ Ω |∇f |2dx− 1 4 ∫ Ω f2 δ2 dx ] .”
Hardy-type inequalities related to degenerate elliptic differential operators (2006, background): “Furthermore, our main results represent a generalization of some results contained in [4, 5].”
On a question of Brezis and Marcus (2006, background): “The proof we present is slightly simpler than in [1].”
The extremal solution of a boundary reaction problem (2008, result): “We will give here a different proof of this result which offers a sharper version, analogous to improvements of (8) found by Brezis and Vázquez [4] or Vázquez and Zuazua [22] (see also [2, 4, 12, 17, 22] for other improved versions of the Hardy inequality (8)) : Theorem 1.”
Optimizing Improved Hardy Inequalities (2002, result): “the right hand side; see, e.g, [BM], [BMS], [ BFT ], [FHT], [GGM], [VZ].”
Sharp Hardy–Sobolev inequalities (2004, background): “Then the following inequality is true ([1], Theorem 6.”