An extremal problem for volume functionals

R. R. Salimov1, B. A. Klishchuk2
1) Institute of Mathematics of the NAS of Ukraine; 2) Institute of Mathematics of the NAS of Ukraine
We consider the class of ring $Q$-homeomorphisms with respect to p-modulus in $\Bbb R^{n}$ with $p > n$, and obtain a lower bound for the volume of images of a ball under such mappings. In particular, the following theorem is proved in the paper: Let $D$ be a bounded domain in $\Bbb R^{n}$, $n \geqslant 2$ and let $f\colon D \rightarrow \Bbb R^{n}$ be a ring $Q$-homeomorphism with respect to $p$-modulus at a point $x_0 \in D$ with $p>n$, and the function $Q$ satisfies the condition $ q_{x_{0}}(t) \leqslant q_{0}\,t^{-\alpha},\, q_{0} \in (0, \infty)\,,\, \alpha \in [0, \infty)$ for a.e. $t\in (0, d_0)$, $d_0 = {\rm dist}(x_{0}, \partial D)$. Then for all $r\in (0, d_0)$ the estimate $$ m(fB(x_0, r)) \geqslant \Omega_n\, \left(\frac{p-n}{\alpha+p-n}\right)^{\frac{n(p-1)}{p-n}}q_{0}^{\frac{n}{n-p}} \, r^{\frac{n(\alpha+ p - n)}{p-n}}\,, $$ holds, where $\Omega_n$ is the volume of the unit ball in $\mathbb{R}^{n}$. In addition, in the paper it is solved an extremal problem on minimizing the volume functional of the image of a ball.
ring Q-homeomorphism; p-modulus of a family of curves; quasiconformal mapping; condenser; p-capacity of a condenser; volume functional
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