Theory Of Quasiconformal Mappings Research Articles

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.

Spectral stability estimates of Dirichlet divergence form elliptic operators

We study spectral stability estimates of elliptic operators in divergence form $$-\text {div} [A(w) \nabla g(w)]$$ with the Dirichlet boundary condition in non-Lipschitz domains $${\widetilde{\varOmega }} \subset {\mathbb {C}}$$ . The suggested method is based on the theory of quasiconformal mappings, weighted Sobolev spaces theory and its applications to the Poincare inequalities.

On the boundary behavior of quasiconformal mappings

We discuss some open questions of the theory of quasiconformal mappings related to the field of studies of Professor G. D. Suvorov. The present work is dedicated to his memory.

To the theory of quasiconformal mappings

The open questions of the theory of quasiconformal mappings that are adjacent to the field of studies of Professor Bogdan Bojarski are discussed.

Application of Dzyadyk’s Polynomial Kernels in the Constructive Function Theory

This is a survey of recent results in the constructive theory of functions of complex variable obtained by the author through the application of the theory of Dzyadyk’s kernels combined with the methods and results from modern geometric function theory and the theory of quasiconformal mappings.

On the boundary behavior of quasiconformal mappings

We discuss some open questions of the theory of quasiconformal mappings related to the field of studies of Professor G. D. Suvorov. The present work is dedicated to his memory.

To the theory of quasiconformal mappings

The open questions of the theory of quasiconformal mappings that are adjacent to the field of studies of Professor Bogdan Bojarski are discussed.

Application of Three-Dimensional Quasi-Conformal Mappings to Grid Construction

Two-dimensional conformal mappings are a powerful and elegant tool for solving many mathematical and physical problems. The conformal mapping method is suitable for constructing two-dimensional grids. The quasi-conformal mappings constructed in this paper naturally generalize the application of conformal mappings to grid construction in the three-dimensional case. For a steady irrotational flow of an ideal incompressible fluid, in addition to the velocity potential, two stream functions are introduced. Generalized Cauchy–Riemann conditions from which three-dimensional quasi-conformal mappings follow are presented. The mappings constructed can be represented as a sequence of two-dimensional conformal mappings. Examples of grid construction using the theory of quasi-conformal mappings are given. The best proof of these results is their visualization.

The Loewner equation and Lipschitz graphs

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, H¨older continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic setting and obtain a new and elementary proof that Holder-1/2 driving functions with norm less than 4 generate simple arcs. We also give a sufficient condition for driving functions to generate curves that are graphs of Lipschitz functions.

The Loewner equation and Lipschitz graphs

The proofs of continuity of Loewner traces in the stochastic and in the deterministic settings employ different techniques. In the former setting of the Schramm–Loewner evolution SLE, H¨older continuity of the conformal maps is shown by estimating the derivatives, whereas the latter setting uses the theory of quasiconformal maps. In this note, we adopt the former method to the deterministic setting and obtain a new and elementary proof that Hölder-1/2 driving functions with norm less than 4 generate simple arcs. We also give a sufficient condition for driving functions to generate curves that are graphs of Lipschitz functions.

On Lebesgue constants for interpolation points on a quasiconformal arc

Using the theory of quasiconformal mappings, we simplify the proof of the recent result by Taylor and Totik (2010) on the behavior of the Lebesgue constants for interpolation points on a compact set in the complex plane.

Nicolas-Auguste Tissot: a link between cartography and quasiconformal theory

Nicolas-Auguste Tissot (1824–1897) published a series of papers on cartography in which he introduced a tool which became known later on, among geographers, under the name of the Tissot indicatrix. This tool was broadly used during the twentieth century in the theory and in the practical aspects of the drawing of geographical maps. The Tissot indicatrix is a graphical representation of a field of ellipses on a map that describes its distortion. Tissot studied extensively, from a mathematical viewpoint, the distortion of mappings from the sphere onto the Euclidean plane that are used in drawing geographical maps, and more generally he developed a theory for the distortion of mappings between general surfaces. His ideas are at the heart of the work on quasiconformal mappings that was developed several decades after him by Grotzsch, Lavrentieff, Ahlfors and Teichmuller. Grotzsch mentions the work of Tissot, and he uses the terminology related to his name (in particular, Grotzsch uses the Tissot indicatrix). Teichmuller mentions the name of Tissot in a historical section in one of his fundamental papers where he claims that quasiconformal mappings were used by geographers, but without giving any hint about the nature of Tissot’s work. The name of Tissot is missing from all the historical surveys on quasiconformal mappings. In the present paper, we report on this work of Tissot. We shall mention some related works on cartography, on the differential geometry of surfaces, and on the theory of quasiconformal mappings. This will place Tissot’s work in its proper context.

Non-autonomous functionals, borderline cases and related function classes

We consider a class of non-autonomous functionals characterised by the fact that the energy density changes its ellipticity and growth properties according to the point, and prove some regularity results for related minimisers. These results are the borderline counterpart of analogous ones previously derived for non-autonomous functionals with (p, q)-growth. We also discuss similar functionals related to Musielak-Orlicz spaces in which basic properties like density of smooth functions, boundedness of maximal and integral operators, and validity of Sobolev type inequalities naturally relate to the assumptions needed to prove regularity of minima. 1. Almost fifty years of degenerate operators in Russia In 1967 a seminal paper [60] of Ural’tseva appeared, featuring the proof of the C-nature of energy solutions to the degenerate equation (1.1) − div(|Du|p−2Du) = 0 . The one on the left hand side is nowadays very well known as the p-Laplacean operator. This operator is relevant in a large number of situations as for instance in the Calculus of Variations, in Geometric Analysis, in the theory of quasiconformal mappings, in the modelling of non-Newtonian fluids. Ural’tseva herself, by exhibiting a counterexample, showed that the regularity of solutions does not go beyond Holder continuity of the gradient, for some exponent β ∈ (0, 1). The proof the Holder gradient continuity result also appears in the second, yet untranslated, edition of the classical book [41]. Ural’tseva’s fundamental result is at the origin of a huge literature, up to the point that it is nowadays hard to find another single nonlinear operator that has attracted so much attention as long as the elliptic regularity theory is concerned. We quote here the important paper of Uhlenbeck [59], where Ural’tseva’s result has been extended to the vectorial case, and the papers [23, 29, 42, 47], where different proofs and extensions to equations with coefficients have been given. The equation appearing in (1.1) is the Euler-Lagrange equation of the functional (1.2) w 7→ ∫ |Dw| dx , p > 1 and, in fact, several of the results and techniques coming from the analysis of (1.1) have been eventually found to be useful in the Calculus of Variations. Starting from the eighties, new models and functionals related to the one in (1.2) were developed by Zhikov [62, 63, 64, 65, 66], together with a group of Russian mathematicians, in order to describe the behaviour of strongly anisotropic materials in the context of homogenisation and nonlinear elasticity. These functionals revealed to be important 1 2 BARONI, COLOMBO, AND MINGIONE also in the study of duality theory and in the context of the Lavrentiev phenomenon. They are non-autonomous functionals of the form

On a Hölder Constant in the Theory of Quasiconformal Mappings

A K-quasiconformal selfmap of the unit disk with identity boundary values satisfies the H\"older estimate $|f(z)-f(w)| \leq 4^{1-1/K} |z-w|^{1/K}$. The constant $4^{1-\frac{1}{K}}$ is sharp.

Extremal Length Decomposition and Domain Constants for Finitely Connected Domains

This paper is concerned with the study of quasi-extremal distance domains, a class of domains introduced by Gehring and Martio in connection with the theory of quasiconformal mappings. We obtain a sharp upper bound for the quasi-extremal distance constant $$M(\Omega )$$ of a finitely connected planar domain in terms of local boundary dilatation of its boundary components. For the proof of the main theorem, several independently interesting results are also established. One of them is a decomposition lemma about the extremal length of a curve family.

On generalized quasiconformal mappings

We study homeomorphisms satisfying some generalized modular inequalities for which we give boundary extension theorems, equicontinuity results and we study the limit mappings of such homeomorphisms. In this way we extend many known facts from the theory of quasiconformal mappings. The main instrument used in this article is the modular method.

Bi-Lipschitz maps in Q-regular Loewner spaces

By applying the theory of quasiconformal maps in measure metric spaces that was introduced by Heinonen-Koskela, we characterize bi-Lipschitz maps by modulus inequalities of rings and maximal, minimal derivatives in Q-regular Loewner spaces. Meanwhile the sufficient and necessary conditions for quasiconformal maps to become bi-Lipschitz maps are also obtained. These results generalize Rohde’s theorem in ℝn and improve Balogh’s corresponding results in Carnot groups.

Pucci's conjecture and the Alexandrov inequality for elliptic PDEs in the plane

The inequality of Alexandrov, Bakel’man and Pucci is a basic tool in the theory of linear elliptic partial differential equations (PDEs) which are not in divergence form as well as in the more general theory of nonlinear elliptic PDEs. Here, in two dimensions, we prove the sharp form of the maximum principle as conjectured by Pucci in 1966, give sharp forms of removable singularity results and prove a number of results for the degenerate elliptic setting. These results make use of the substantial recent advances in the planar theory of quasiconformal mappings. ∗Astala was supported in part by the Academy of Finland, projects 34082 and 41933, Iwaniec by the National Science Foundation grants DMS-0301582 and DMS-0244297, Martin by the N.Z. Marsden Fund and the N.Z. Royal Society (James Cook Fellowship) AMS (1991) Classification. Primary 30C60, 35J15, 35J70

On a functional equation arising in the theory of quasiconformal mappings

We solve functional equations of the form $$k(x,h) = \frac{{f(x + h) - f(x)}}{{f(x) - f(x - h)}},\quad x \in {\mathbb{R}},\;h > 0,$$ in the class of injective functions \(f:{\mathbb{R}}: \to {\mathbb{R}},\) provided \(k:{\mathbb{R}} \times (0,\infty ) \to {\mathbb{R}}\) is a product of two functions, each of them of one variable.

Estimates for the Discrepancy of a Signed Measure Using Its Energy Norm

W. Kleiner established (1964, Ann. Polon. Math. 14 , 117–130) for smooth curves and arcs an estimate for the discrepancy of a signed measure by using its energy norm. We extend this result to quasiconformal curves and arcs. The proof uses the theory of quasiconformal mappings and condenser theory. In a first step, the discrepancy of a signed measure can be estimated from above in terms of its energy norm and the capacity of a special condenser. This result is valid on every Jordan curve. Examples show the sharpness of the results from various points of view.