An extremal problem for volume functionals

R. R. Salimov; B. A. Klishchuk

doi:10.15330/ms.50.1.36-43

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.

1. B.V. Bojarski, Homeomorphic solutions to Beltrami systems, Dokl. Akad. Nauk SSSR, 102 (1955), 661–664 (in Russian).

2. F.W. Gehring, E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 388 (1966), 1–15.

3. K. Astala, Area distortion under quasiconformal mappings, Acta Math., 173 (1994), №1, 37–60, doi: 10.1007/BF02392568.

4. A. Eremenko, D. H. Hamilton, On the area distortion by quasiconformal mappings, Proc. Amer. Math. Soc., 123 (1995), 2793–2797, doi: 10.1090/S0002-9939-1995-1283548-8

5. M.A. Lavrentiev, The variational method in boundary-value problems for systems of equations of elliptic type, Moscow: Izd-vo AN SSSR, 1962 (in Russian).

6. B. Bojarski, V. Gutlyanskii, O. Martio, V. Ryazanov, Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane, EMS Tracts in Mathematics, 19. Zurich, 2013, x+205 pp.

7. T.V. Lomako, R.R. Salimov, To the theory of extremal problems, Zb. Pr. Inst. Mat. NAN Ukr., 7 (2010), №2, 264–269 (in Russian).

8. R.R. Salimov Lower estimates of p-modulus and mappings of Sobolev’s class, Algebra i analiz, 26 (2014), №6, 143–171 (in Russian). Engl. transl.: St. Petersbg. Math. J., 26 (2015), №6, 965–984, doi: 10.1090%2Fspmj%2F1370

9. V.I. Kruglikov, Capacity of condensers and spatial mappings quasiconformal in the mean, Matem. sborn., 130 (1986), №2, 185–206 (in Russian). Engl. transl.: Math. USSR-Sbornik, 58 (1987), №1, 185–205, doi: 10.1070%2FSM1987v058n01ABEH003099

10. R.R. Salimov, Estimation of the measure of the image of the ball, Sibir. Matem. Zh., 53 (2012), №4, 920-930 (in Russian). Engl. transl.: Sib. Math. J. 53 (2012), №4, 739-747, doi: 10.1134/S0037446612040155

11. R.R. Salimov, B.A. Klishchuk, The extremal problem for the area of an image of a disc, Dopov. NANU, (2016), №10, 22-27 (in Russian).

12. B.A. Klishchuk, R.R. Salimov, Lower bounds for the area of the image of a circle, Ufa Math. J., 9 (2017), №2, 55-61, doi: 10.13108%2F2017-9-2-55

13. R.R. Salimov, B.A. Klishchuk, Extremal problem for the areas of the images of a disks, Zap. Nauchn. Sem. POMI, 456 (2017), 160-171 (in Russian). Enlg. transl.: J. Math. Sci. 234 (2018), №3, 373-380, doi: 10.1007/s10958-018-4015-6

14. O. Martio, S. Rickman, and J. Vaisala, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1. Math., 448 (1969), 1-40.

15. V.A. Shlyk, The equality between p-capacity and p-modulus, Sibir. Matem. Zh., 34 (1993), №6, 1196.1200 (in Russian). Engl. transl.: Sib. Mat. J., 34 (1993), №6, 1196-1200, doi: 10.1007%2FBF00973485

16. S. Saks, Theory of the integral, G. E. Stechert & Co., New York (1937).

17. F.W. Gehring, Lipschitz mappings and the p-capacity of ring in n-space, Advances in the theory of Riemann surfaces (Proc. Conf. Stonybrook, N.Y., 1969), Ann. Math. Studies, 66 (1971), 175-193.

	An extremal problem for volume functionals
Author	R. R. Salimov¹, B. A. Klishchuk² ruslan.salimov1@gmail.com¹, kban1988@gmail.com² 1) Institute of Mathematics of the NAS of Ukraine; 2) Institute of Mathematics of the NAS of Ukraine
Abstract	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.
Keywords	ring Q-homeomorphism; p-modulus of a family of curves; quasiconformal mapping; condenser; p-capacity of a condenser; volume functional
DOI	doi:10.15330/ms.50.1.36-43
Reference	1. B.V. Bojarski, Homeomorphic solutions to Beltrami systems, Dokl. Akad. Nauk SSSR, 102 (1955), 661–664 (in Russian). 2. F.W. Gehring, E. Reich, Area distortion under quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math., 388 (1966), 1–15. 3. K. Astala, Area distortion under quasiconformal mappings, Acta Math., 173 (1994), №1, 37–60, doi: 10.1007/BF02392568. 4. A. Eremenko, D. H. Hamilton, On the area distortion by quasiconformal mappings, Proc. Amer. Math. Soc., 123 (1995), 2793–2797, doi: 10.1090/S0002-9939-1995-1283548-8 5. M.A. Lavrentiev, The variational method in boundary-value problems for systems of equations of elliptic type, Moscow: Izd-vo AN SSSR, 1962 (in Russian). 6. B. Bojarski, V. Gutlyanskii, O. Martio, V. Ryazanov, Infinitesimal geometry of quasiconformal and bi-Lipschitz mappings in the plane, EMS Tracts in Mathematics, 19. Zurich, 2013, x+205 pp. 7. T.V. Lomako, R.R. Salimov, To the theory of extremal problems, Zb. Pr. Inst. Mat. NAN Ukr., 7 (2010), №2, 264–269 (in Russian). 8. R.R. Salimov Lower estimates of p-modulus and mappings of Sobolev’s class, Algebra i analiz, 26 (2014), №6, 143–171 (in Russian). Engl. transl.: St. Petersbg. Math. J., 26 (2015), №6, 965–984, doi: 10.1090%2Fspmj%2F1370 9. V.I. Kruglikov, Capacity of condensers and spatial mappings quasiconformal in the mean, Matem. sborn., 130 (1986), №2, 185–206 (in Russian). Engl. transl.: Math. USSR-Sbornik, 58 (1987), №1, 185–205, doi: 10.1070%2FSM1987v058n01ABEH003099 10. R.R. Salimov, Estimation of the measure of the image of the ball, Sibir. Matem. Zh., 53 (2012), №4, 920-930 (in Russian). Engl. transl.: Sib. Math. J. 53 (2012), №4, 739-747, doi: 10.1134/S0037446612040155 11. R.R. Salimov, B.A. Klishchuk, The extremal problem for the area of an image of a disc, Dopov. NANU, (2016), №10, 22-27 (in Russian). 12. B.A. Klishchuk, R.R. Salimov, Lower bounds for the area of the image of a circle, Ufa Math. J., 9 (2017), №2, 55-61, doi: 10.13108%2F2017-9-2-55 13. R.R. Salimov, B.A. Klishchuk, Extremal problem for the areas of the images of a disks, Zap. Nauchn. Sem. POMI, 456 (2017), 160-171 (in Russian). Enlg. transl.: J. Math. Sci. 234 (2018), №3, 373-380, doi: 10.1007/s10958-018-4015-6 14. O. Martio, S. Rickman, and J. Vaisala, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A1. Math., 448 (1969), 1-40. 15. V.A. Shlyk, The equality between p-capacity and p-modulus, Sibir. Matem. Zh., 34 (1993), №6, 1196.1200 (in Russian). Engl. transl.: Sib. Mat. J., 34 (1993), №6, 1196-1200, doi: 10.1007%2FBF00973485 16. S. Saks, Theory of the integral, G. E. Stechert & Co., New York (1937). 17. F.W. Gehring, Lipschitz mappings and the p-capacity of ring in n-space, Advances in the theory of Riemann surfaces (Proc. Conf. Stonybrook, N.Y., 1969), Ann. Math. Studies, 66 (1971), 175-193.
Pages	36-43
Volume	50
Issue	1
Year	2018
Journal	Matematychni Studii
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