An extremal problem for volume functionals
Journal Title: Математичні Студії - Year 2018, Vol 50, Issue 1
Abstract
We consider the class of ring Q-homeomorphisms with respect to p-modulus in Rn 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 Rn, n⩾2 and let f:D→Rn be a ring Q-homeomorphism with respect to p-modulus at a point x0∈D with p>n, and the function Q satisfies the condition qx0(t)⩽q0t−α,q0∈(0,∞),α∈[0,∞) for a.e. t∈(0,d0), d0=dist(x0,∂D). Then for all r∈(0,d0) the estimate m(fB(x0,r))⩾Ωn(p−nα+p−n)n(p−1)p−nqnn−p0rn(α+p−n)p−n, holds, where Ωn is the volume of the unit ball in Rn. In addition, in the paper it is solved an extremal problem on minimizing the volume functional of the image of a ball.
Authors and Affiliations
R. R. Salimov, B. A. Klishchuk
Keywords
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- EP ID EP436192
- DOI 10.15330/ms.50.1.36-43
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