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

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 ...