Isoperimetric inequalities for the least harmonic majorant of $\vert x\vert ^ p$

Abstract:

Let $D$ be an open set in the $d$-dimensional Euclidean space ${{\mathbf {R}}^d}$ containing the origin $0$ and let ${h^{(p)}}(x,D)$ be the least harmonic majorant of $|x{|^p}$ in $D$, where $0 < p < \infty$ if $d \geqslant 2$ and $1 \leqslant p < \infty$ if $d = 1$. We shall be concerned with the following isoperimetric inequalities ${h^{(p)}}{(0,D)^{1/p}} \leqslant cr(D)$, where $r(D)$ denotes the volume radius of $D$, namely, a ball with radius $r(D)$ has the same volume as $D$ has and $c$ is a constant dependent on $d$ and $p$ but independent of $D$. We fix $d$ and denote by $c(p)$ the infimum of such constants $c$. As a function of $p$, $c(p)$ is nondecreasing and satisfies $c(p) \geqslant 1$. We shall show (1) there are positive constants ${C_1}$ and ${C_2}$ such that ${C_1}{p^{(d - 1)/d}} \leqslant c(p) \leqslant {C_2}{p^{(d - 1)/d}}$ for $p \geqslant 1$, (2) $c(p) = 1$ if $p \leqslant d + {2^{1 - d}}$. Many estimations of ${h^{(p)}}(0,D)$ and their applications are also given.