A reverse Holder inequality for extremal Sobolev functions

Abstract

Let $n\ge 2$, let $\Omega \subset {\mathbf{R}}^n$ be a bounded domain with smooth boundary, and let $1\le p\le 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is $${\mathcal{C}}_p(\Omega )=\inf \left\{\frac{{\int }_{\Omega }|\nabla v{|}^2}{{\left({\int }_{\Omega }|v{|}^p\right)}^{2/p}}\right\}=\frac{{\int }_{\Omega }|\nabla u{|}^2}{{\left({\int }_{\Omega }|u{|}^p\right)}^{2/p}}.$$ Our inequality has the form $\Vert u{\Vert }_{L^p}\ge K\Vert u{\Vert }_{L^q}$ for any $q>p$, where $K$ depends only on $n$, $p$, $q$, and ${\mathcal{C}}_p(\Omega )$. This result generalizes work of Chiti, regarding the first Dirichlet eigenfunction of the Laplacian, and of van den Berg, regarding the torsion function.

10 pages

References in corpus (2)

• Two isoperimetric inequalities for the Sobolev constant
• An isoperimetric inequality for extremal Sobolev functions