Isoperimetric inequalities for $p$-conductance

A. L. Fedorov

Matematičeskie zametki, Tome 11 (1972) no. 3, pp. 275-282

Résumé

An $n$-dimensional domain $K$ is considered with boundary $\partial K=K_0\cup K_1\cup K_2$ such that the closure $\overline{K}$ is the image of a cylinder $B=S\times[0,1]$ ($S$ is a closed $(n-1)$-dimensional cell) under a one-one Lipschitz map. For the $p$-conductance of the domain $K$, defined by the equation $$ c_p(K)=\inf_{U(K)}\int_K|\nabla f|^pdx\qquad(p>1), $$ where $U(K)=\{f(x):f\in W_p^1(K)\cap C(\overline{K}), f=1 \text{ на } K_1, f=0 \text{ на } K_0\}$, the isoperimetric inequality $c_p(K)\leqslant V/r^p$ is established. Here $V$ is the $n$-dimensional volume of the domain $K$, $r$ is the shortest distance between $K_0$ and $K_1$, measured in $K$. Equality is achieved on the right cylinder.