1. Hardy's inequality in a limiting case on general bounded domains.
- Author
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Byeon, Jaeyoung and Takahashi, Futoshi
- Subjects
- *
RADIUS (Geometry) , *MATHEMATICAL equivalence , *BLOWING up (Algebraic geometry) - Abstract
In this paper, we study Hardy's inequality in a limiting case: ∫ Ω | ∇ u | N d x ≥ C N (Ω) ∫ Ω | u (x) | N | x | N (log R | x |) N d x for functions u ∈ W 0 1 , N (Ω) , where Ω is a bounded domain in ℝ N with R = sup x ∈ Ω | x |. We study the attainability of the best constant C N (Ω) in several cases. We provide sufficient conditions that assure C N (Ω) > C N (B R) and C N (Ω) is attained, here B R is the N -dimensional ball with center the origin and radius R. Also, we provide an example of Ω ⊂ ℝ 2 such that C 2 (Ω) > C 2 (B R) = 1 / 4 and C 2 (Ω) is not attained. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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