On one extremal problem for nonlinear Cauchy-Riemann-Beltrami systems

The study of nonlinear Cauchy--Riemann--Beltrami systems is conditioned study of certain problems of hydrodynamics and gas dynamics, in which there is an inhomogeneity of media and a certain singularity. The paper considers a nonlinear Cauchy--Riemann--Beltrami type system in the polar coordinate system in which the radial derivative is expressed through the complex coefficient, the angular derivative and its m-degree module. In particular, if m is equal to zero, then this system of equations is reduced to the ordinary linear system of Beltrami equations. Note that general first-order systems were used by M.А. Lavrentyev to define quasiconformal mappings on the plane, see \cite{L}. The problem of area distortion under quasi-conformal mappings is due to the work of B. Boyarsky, see \cite{Bo}. For the first time, the upper estimate of the area of the disk image under quasi-conformal mappings was obtained by M.А. Lavrentyev, see \cite{L}. A refinement of the Lavrentyev inequality in terms of the angular dilatation was obtained in the monograph \cite{BGMR}, see Proposition 3.7. In the present paper, it is found an exact upper estimate of the area of the image of the disk, which is analogous to the known result by Lavrentyev. Also, we find here a mapping on which the estimate is achieved. Thus, the work solves the extreme problem for the area functional of the image of disks under a certain class of regular homeomorphic solutions of nonlinear systems of the Cauchy--Riemann--Beltrami type with generalized derivatives integrated with a square. The work uses p-angular dilatation. In the conformal case, angular dilatation is important in the theory of quasi-conformal mappings and nondegenerate Beltrami equations. Proof of the main result of the article is based on the differential relation for the area function of the image of disks of arbitrary radii, which was established in the previous work of the authors for regular homeomorphisms with Luzin's N-property.

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On Hölder continuity of solutions of the Beltrami equations on the boundary

Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine ◽

10.37069/1683-4720-2018-32-11 ◽

2018 ◽

Vol 32 ◽

pp. 104-114

Author(s):

Vladimir Ryazanov ◽

Ruslan Salimov

Keyword(s):

Boundary Value Problems ◽

Unit Disk ◽

Sobolev Class ◽

Hilbert Problem ◽

Boundary Value ◽

Beltrami Equation ◽

Logarithmic Capacity ◽

Complex Coefficient ◽

Beltrami Equations ◽

Riemann Hilbert Problem

In the present paper, it is found conditions on the complex coefficient of the Beltrami equations with the degeneration of the uniform ellipticity in the unit disk under which their generalized homeomorphic solutions are continuous by Hölder on the boundary. These results can be applied to the investigations of various boundary value problems for the Beltrami equations. In a series of recent papers, under the study of the boundary value problems of Dirichlet, Hilbert, Neumann, Poincare and Riemann with arbitrary measurable boundary data for the Beltrami equations as well as for the generalizations of the Laplace equation in anisotropic and inhomogeneous media, it was applied the logarithmic capacity, see e.g. Gutlyanskii V., Ryazanov V., Yefimushkin A. On the boundary value problems for quasiconformal functions in the plane // Ukr. Mat. Visn. - 2015. - 12, no. 3. - P. 363-389; transl. in J. Math. Sci. (N.Y.) - 2016. - 214, no. 2. - P. 200-219; Gutlyanskii V., Ryazanov V., Yefimushkin A. On a new approach to the study of plane boundary-value problems // Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki. - 2017. - No. 4. - P. 12-18; Yefimushkin A. On Neumann and Poincare Problems in A-harmonic Analysis // Advances in Analysis. - 2016. - 1, no. 2. - P. 114-120; Efimushkin A., Ryazanov V. On the Riemann-Hilbert problem for the Beltrami equations in quasidisks // Ukr. Mat. Visn. - 2015. - 12, no. 2. - P. 190–209; transl. in J. Math. Sci. (N.Y.) - 2015. - 211, no. 5. - P. 646–659; Yefimushkin A., Ryazanov V. On the Riemann–Hilbert Problem for the Beltrami Equations // Contemp. Math. - 2016. - 667. - P. 299-316; Gutlyanskii V., Ryazanov V., Yakubov E., Yefimushkin A. On Hilbert problem for Beltrami equation in quasihyperbolic domains // ArXiv.org: 1807.09578v3 [math.CV] 1 Nov 2018, 28 pp. As well known, the logarithmic capacity of a set coincides with the so-called transfinite diameter of the set. This geometric characteristic implies that sets of logarithmic capacity zero and, as a consequence, measurable functions with respect to logarithmic capacity are invariant under mappings that are continuous by Hölder. That circumstance is a motivation of our research. Let $D$ be a domain in the complex plane $\mathbb C$ and let $\mu: D\to\mathbb C$ be a measurable function with $ |\mu(z)| \lt 1 $ a.e. The equation of the form $f_{\bar{z}}\ =\ \mu(z) f_z $ where $ f_{\bar z}={\bar\partial}f=(f_x+if_y)/2 $, $f_{z}=\partial f=(f_x-if_y)/2$, $z=x+iy$, $ f_x $ and $ f_y $ are partial derivatives of the function $f$ in $x$ and $y$, respectively, is said to be a Beltrami equation. The function $\mu$ is called its complex coefficient, and $ K_{\mu}(z)=\frac{1+|\mu(z)|}{1-|\mu(z)|}$ is called its dilatation quotient. The Beltrami equation is said to be degenerate if ${\rm ess}\,{\rm sup}\,K_{\mu}(z)=\infty$. The existence of homeomorphic solutions in the Sobolev class $W^{1,1}_{\rm loc}$ has been recently established for many degenerate Beltrami equations under the corresponding conditions on the dilatation quotient $K_{\mu}$, see e.g. the monograph Gutlyanskii V., Ryazanov V., Srebro U., Yakubov E. The Beltrami equation. A geometric approach. Developments in Mathematics, 26. Springer, New York, 2012 and the further references therein. The main theorem of the paper, Theorem 1, states that a homeomorphic solution $ f:\mathbb D\to\mathbb D $ in the Sobolev class $ W^{1,1}_{\rm loc} $ of the Beltrami equation in the unit disk $\mathbb D$ has a homeomorphic extension to the boundary that is Hölder continuous if $K_{\mu}\in L^1(\Bbb D)$ and, for some $\varepsilon_0\in(0,1)$ and $C\in[1,\infty)$, $$ \sup\limits_{\varepsilon\in(0,\varepsilon_0)} \int_{\mathbb D\cap D(\zeta,\varepsilon)}K_{\mu}(z) dm(z) \lt C \qquad \forall \zeta \in \partial \mathbb{D} $$ where $D(\zeta,\varepsilon)=\left\{z\in{\Bbb C}: |z-\zeta| \lt \varepsilon\right\}$.

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Ultrastructural characteristics of sporidia of Ustilago

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100155906 ◽

1989 ◽

Vol 47 ◽

pp. 786-787

Author(s):

Patricia N. Hackney

Keyword(s):

Electron Microscope ◽

Type System ◽

Tube Formation ◽

High Voltage Electron Microscopy ◽

Voltage Electron ◽

University Of Wisconsin ◽

Transmission Electron ◽

Ustilago Hordei ◽

Silene Alba ◽

The University

Ustilago hordei and Ustilago violacea are yeast-like basidiomycete pathogens ofHordeum vulgare and Silene alba respectively. The mating type system in both species of Ustilago is bipolar, with alleles, A,a, (U.hordei) and a1, a2 (U.violacea) at a single locus. Haploid sporidia maintain the asexual phase by budding, while the sexual phase is initiated by conjugation tube formation between the mating types during budding and conjugation.For observation of budding, sporidia were prepared by culturing the four types on YEG (yeast extract glucose) broth for 24 hours. After centrifugation at 5000g cells were either left unmated or mated in a1/a2,A/a combinations. The sporidia were then mixed 1:1 with 4% agar and the resulting 1mm cubes fixed in 8% gluteraldehyde and post fixed in osmium tetroxide. After dehydration and embedding cubes were thin sectioned with a LKB ultratome and photographed in a Zeiss 9s transmission electron microscope or in an AE1 electron microscope of MK11 1MEV at the High Voltage Electron Microscopy Center of the University of Wisconsin-Madison.

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