Upper and lower bounds for the optimal constant in the extended Sobolev inequality. Derivation and numerical results

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 λ_n,α for u ∈ H¹(ℝⁿ) of the functional Λ_n,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 λ_n,α for all q _ = max(0,-q) ∈ L^p(ℝⁿ), for n = 1, 1 ≤ p < ∞, and n ≥ 2, n/2 < p < ∞, and the norm ∥q _ ∥p.