Print Email Facebook Twitter Upper and lower bounds for the optimal constant in the extended Sobolev inequality. Derivation and numerical results Title Upper and lower bounds for the optimal constant in the extended Sobolev inequality. Derivation and numerical results Author Nasibov, Sh M. (Baku State University) Veling, E.J.M. (TU Delft Water Resources) Date 2019 Abstract We prove and give numerical results for two lower bounds and eleven upper bounds to the optimal constant k0 = k0(n,α) in the inequality ∥u∥2n/(n-2α) ≤ k0 ∥∇u∥α2 ∥u∥1-α 2, u ∈ H1(ℝn), for n = 1, 0 < α ≤ 1/2, and n ≥ 2, 0 < α < 1. This constant k0 is the reciprocal of the infimum λn,α for u ∈ H1(ℝn) of the functional Λn,α = ∥∇u∥α2 ∥u∥1-α 2/∥u∥2n/(n-2α), u ∈ H1(ℝn), where for n = 1, 0 < α ≤ 1/2, and for n ≥ 2, 0 < α < 1. The lowest point in the point spectrum of the Schrödinger operator τ = -Δ+q on ℝn with the real-valued potential q can be expressed in λn,α for all q _ = max(0,-q) ∈ Lp(ℝn), for n = 1, 1 ≤ p < ∞, and n ≥ 2, n/2 < p < ∞, and the norm ∥q _ ∥p. Subject Lower boundOptimal constantSobolev inequalityUpper bound To reference this document use: http://resolver.tudelft.nl/uuid:cc6471d5-cdce-48b4-abd3-521bd31e227f DOI https://doi.org/10.7153/jmi-2019-13-52 ISSN 1846-579X Source Journal of Mathematical Inequalities, 13 (3), 753-778 Part of collection Institutional Repository Document type journal article Rights © 2019 Sh M. Nasibov, E.J.M. Veling Files PDF jmi_13_52.pdf 441.57 KB Close viewer /islandora/object/uuid:cc6471d5-cdce-48b4-abd3-521bd31e227f/datastream/OBJ/view