Abstract:
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.
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