TY - JOUR
T1 - Upper and lower bounds for the optimal constant in the extended Sobolev inequality. Derivation and numerical results
AU - Nasibov, Sh M.
AU - Veling, E. J.M.
PY - 2019
Y1 - 2019
N2 - 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.
AB - 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.
KW - Lower bound
KW - Optimal constant
KW - Sobolev inequality
KW - Upper bound
UR - http://www.scopus.com/inward/record.url?scp=85073221472&partnerID=8YFLogxK
U2 - 10.7153/jmi-2019-13-52
DO - 10.7153/jmi-2019-13-52
M3 - Article
AN - SCOPUS:85073221472
SN - 1846-579X
VL - 13
SP - 753
EP - 778
JO - Journal of Mathematical Inequalities
JF - Journal of Mathematical Inequalities
IS - 3
ER -