A Reverse Hölder Inequality for Extremal Sobolev Functions

Abstract

Let n ≥ 2, let Ω ⊂ ℝⁿ be a bounded domain with $\mathcal {C}^{1}$ boundary, and let $1 \leq p < \frac {2n}{n-2}$ (simply p ≥ 1 if n = 2). The well-known Sobolev imbedding theorem and Rellich compactness implies that

$$\mathcal{C}_{p}(\Omega) = \inf \left \{ \frac{\int_{\Omega} |\nabla f|^{2} dm}{\left ( \int_{\Omega} |f|^{p} dm \right )^{2/p}} : f \in W^{1,2}_{0}(\Omega) , f \not \equiv 0\right \} $$

is a finite, positive number, and the infimum is achieved by a nontrivial extremal function u, which one can assume is positive inside Ω. We prove that, for 1≤p ≤ 2 and for every q>p, there exists $K= K(n, p, q, \mathcal {C}_{p}(\Omega ))>0$ such that ∥u∥ _{Lp (Ω)} ≥ K∥u∥ _{Lq (Ω)}. This inequality, which reverses the classical Hölder inequality, mirrors results of G. Chiti for the first Dirichlet eigenfunction of the Laplacian and of M. van den Berg for the torsion function.