A reverse Holder inequality for extremal Sobolev functions

الأصل · EN

Let n ≥ 2, let Ω⊂ Rⁿ be a bounded domain with smooth boundary, and let 1 ≤ p ≤ 2. We prove a reverse-Holder inequality for functions u realizing the best constant in the Sobolev inequality, that is Cₚ(Ω) = { ∫Ω|∇ v|² (∫Ω|v|ᵖ)²/ᵖ } = ∫Ω|∇ u|² (∫Ω|u|ᵖ)²/ᵖ. Our inequality has the form u ₗₚ ≥ K u ₗq for any q > p, where K depends only on n, p, q, and Cₚ(Ω). This result generalizes work of Chiti, regarding the first Dirichlet eigenfunction of the Laplacian, and of van den Berg, regarding the torsion function.