p-capacity vs surface-area

Abstract

This paper is devoted to exploring the relationship between the $[1,n)\ni p$-capacity and the surface-area in $\mathbb R^{n\ge 2}$ which especially shows: if $Ω\subset\mathbb R^n$ is a convex, compact, smooth set with its interior $Ω^\circ\not=\emptyset$ and the mean curvature $H(\partialΩ,\cdot)>0$ of its boundary $\partialΩ$ then $$ \left(\frac{n(p-1)}{p(n-1)}\right)^{p-1}\le\frac{\left(\frac{\hbox{cap}_p(Ω)}{\big(\frac{p-1}{n-p}\big)^{1-p}σ_{n-1}}\right)}{\left(\frac{\hbox{area}(\partialΩ)}{σ_{n-1}}\right)^\frac{n-p}{n-1}}\le\left(\sqrt[n-1]{\int_{\partialΩ}\big(H(\partialΩ,\cdot)\big)^{n-1}\frac{dσ(\cdot)}{σ_{n-1}}}\right)^{p-1}\quad\forall\quad p\in (1,n) $$ whose limits $1\leftarrow p\ \&\ p\rightarrow n$ imply $$ 1=\frac{cap_1(Ω)}{\hbox{area}(\partialΩ)}\ \ \& \ \int_{\partialΩ}\big(H(\partialΩ,\cdot)\big)^{n-1}\frac{dσ(\cdot)}{σ_{n-1}}\ge 1, $$ thereby not only discovering that the new best known constant is roughly half as far from the one conjectured by Pólya-Szegö in \cite[(2)]{P} but also extending the Pólya-Szegö inequality in \cite[(5)]{P}, with both the conjecture and the inequality being stated for the electrostatic capacity of a convex solid in $\mathbb R^3$.

13 pages