Quasiconformal Research Articles

Local behavior of homeomorphisms with a finite mean over spheres

It is well known that the modulus method is one of the most powerful tools for studying mappings. Distortion estimates of the modulus of paths families are established in many known classes, in particular, the modulus does not change under conformal mappings, is finitely distorted under qu\-a\-si\-con\-for\-mal mappings, at the same time, its behavior under mappings with finite distortion depends on the dilatation coefficient. One common case is the study of mappings for which this coefficient is integrable in the domain. In the context of our research, this case has been studied in detail in our previous publications and its consideration has mostly been completed. In particular, we obtained results on the local, boundary, and global behavior of homeomorphisms, the inverse of which satisfy the weight Poletsky inequality, provided that the corresponding majorant is integrable. In contrast, the focus in this paper is on mappings for which a similar inequality may contain non integrable weights. Study of the situation of non integrable majorants, in turn, is associated with the specific behavior of the weight modulus of the annulus, which is achieved on a certain function and up to constant is equal to $(n-1)$-degree of the Lehto integral. To the same extent, these results are also related to finding the extremal in the weight modulus of the ring. The basic theorem contains the result about equicontinuity of homeomorphisms with the inverse Poletsky inequality, when the corresponding weight has finite integrals on some set of spheres, and the set of corresponding radii of these spheres must have a positive Lebesgue measure. According to Fubini's theorem, the mentioned result summarizes the corresponding statement for any integrable majorants and is fundamental in the sense that it is easy to give examples of non integrable functions with finite integrals by spheres. In addition, since conformal and quasiconformal mappings satisfy the Poletsky inequality with a constant majorant in the forward and inverse directions, the basic theorem may be considered as a generalization of previously known statements in these classes. Note that the main result does not contain any geometric constraints on the definition and image domains of the mappings, in particular, the definition domain is assumed to be arbitrary, and the image domain is supposed to be only a bounded domain in Euclidean $n$-dimensional space. The proof of the main theorem is given by the contradiction, namely, we assume that the statement about equicontinuity of the corresponding family of mappings is incorrect, and we obtain a contradiction to this assumption due to upper and lower estimates of the modulus of families of paths.

On Julia Limiting Directions in Higher Dimensions

For a quasiregular mapping $$f:{\mathbb {R}}^n \rightarrow {\mathbb {R}}^n$$ , with $$n\ge 2$$ , a Julia limiting direction $$\theta \in S^{n-1}$$ arises from a sequence $$(x_n)_{n=1}^{\infty }$$ contained in the Julia set of f, with $$|x_n| \rightarrow \infty $$ and $$x_n/|x_n| \rightarrow \theta $$ . Julia limiting directions have been extensively studied for entire and meromorphic functions in the plane. In this paper, we focus on Julia limiting directions in higher dimensions. First, we give conditions under which every direction is a Julia limiting direction. Our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a sufficient, but not necessary, condition in $${\mathbb {R}}^3$$ for a set $$E\subset S^2$$ to be the set of Julia limiting directions for a quasiregular mapping. The methods here will require showing that certain sectorial domains in $${\mathbb {R}}^3$$ are ambient quasiballs. This is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball $${\mathbb {B}}^3$$ under an ambient quasiconformal mapping of $${\mathbb {R}}^3$$ onto itself.

A note on Väisälä’s problem concerning free quasiconformal mappings

In this paper, we provide partial solutions to a problem raised by Vaisala on local properties of free quasiconformal mappings. In particular, we show that a locally free quasiconformal mapping is a globally free quasiconformal mapping under the condition of locally relative quasisymmetry.

Quasiconformal Harmonic Mappings Between the Unit Ball and a Spatial Domain with C1,α Boundary

We prove the following. If f is a harmonic quasiconformal mapping between the unit ball in $\mathbb {R}^{n}$ and a spatial domain with C1,α boundary, then f is Lipschitz continuous in B. This generalizes some known results for n = 2 and improves some others in higher dimensional case.

Hyperbolically Lipschitz continuity, area distortion and coefficient estimates for -quasiconformal harmonic mappings of unit disk

UDC 517.51 We study the hyperbolically Lipschitz continuity, Euclidean and hyperbolic area distortion theorem, and coefficient estimate for the classes of -quasiconformal harmonic mappings from the unit disk onto itself.

Monotonicity theorems and inequalities for the Hübner function with applications

In this paper, the authors present sharp bounds of the Hübner function M, which is defined by (1.11) and has important applications in the theories of quasiconformal maps and Ramanujan's modular equations, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of M and elementary functions. By these results, several well-known results for M including its bounds and logarithmic inequalities, the explicit quasiconformal Schwarz lemma and the estimates of the solutions to Ramanujan's classical modular equations are remarkably improved. A simpler and more concise proof of the series expansion of M(r) is given, too.

Fredholm Eigenvalues and Quasiconformal Geometry of Polygons

An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory.This survey extends our previous survey of 2005 and presents the new approaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

We construct quasiconformal mappings $f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ for which there is a Borel set $E \subset \mathbb{R}^2 \times \{0\}$ of positive Lebesgue $2$-measure whose image $f(E)$ has Hausdorff $2$-measure zero. This gives a solution to the open problem of inverse absolute continuity of quasiconformal mappings on hypersurfaces, attributed to Gehring. By implication, our result also answers questions of V\"ais\"al\"a and Astala--Bonk--Heinonen.

Duality of moduli in regular metric spaces

F. Gehring and W. Ziemer proved that the p-modulus of the family of paths connecting two continua is dual to the p^*-modulus of the corresponding family of separating hypersurfaces. In this paper we show that a similar result holds in complete Ahlfors-regular metric spaces that support a weak 1-Poincar\'e inequality. As an application we obtain a new characterization for quasiconformal mappings between such spaces.

A direct reconstruction algorithm for the anisotropic inverse conductivity problem based on Calderón’s method in the plane

A direct reconstruction algorithm based on Calderón’s linearization method for the reconstruction of isotropic conductivities is proposed for anisotropic conductivities in two-dimensions. To overcome the non-uniqueness of the anisotropic inverse conductivity problem, the entries of the unperturbed anisotropic tensors are assumed known a priori, and it remains to reconstruct the multiplicative scalar field. The quasi-conformal map in the plane facilitates the Calderón-based approach for anisotropic conductivities. The method is demonstrated on discontinuous radially symmetric conductivities of high and low contrast.

Space quasiconformal composition operators with applications to Neumann eigenvalues

In this article we obtain estimates of Neumann eigenvalues of p-Laplace operators in a large class of space domains satisfying quasihyperbolic boundary conditions. The suggested method is based on composition operators generated by quasiconformal mappings and their applications to Sobolev–Poincare-inequalities. By using a sharp version of the reverse Holder inequality we refine our estimates for quasi-balls, that is, images of balls under quasiconformal mappings of the whole space.

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.

Generalization of numerical quasiconformal mapping methods for geological problems

A method for identifying parameters of the conductivity coefficient of objects is generalized for the case of reconstructing an image of a part of a soil massif from the tomography data of the applied quasipotentials. In this case, without diminishing the generality, the reconstruction of the image is carried out in a fragment of a rectangular medium with local bursts of homogeneity present in it. The general idea of the corresponding algorithm consists in the sequential iterative solution of problems on quasiconformal mappings and identification of the parameters of the conductivity coefficient, with an insufficient amount of data on the values of the flow functions on the “inaccessible” part of the boundary. The image was reconstructed according to the data obtained using a full-range gradient array. The developed approach, in comparison with the existing ones, has a number of advantages that make it possible to increase the accuracy of identification of the conductivity coefficient. Namely, it provides an increase, in a qualitative sense, in the amount of input data, allows avoiding the use of Dirac delta functions when modeling areas of application of potentials and sufficiently flexibly take into account the mathematical aspects of the implementation of a quasiconformal mapping of a finite fragment of a half-plane onto a parametric polygon (domain of a complex quasipotential). The solution of the corresponding problem, in particular, occurs not in a single (fixed) investigated fragment of a rectangular soil massif, but in a number of smaller subdomains of the same shape, in the proposed optimal sequence. This saves machine time significantly. The prospects for further practical implementation of the proposed method follow from its ability to give an approximate result with relatively low costs (financial, time)

Modelling of an explosion impact on a deformable medium with rigid inclusions with quasiconformal mappings methods

The explosive process influence on the environment with the existing impenetrable fixed inclusion is investigated by quasiconformal mappings numerical methods and step-by-step parameterization of the environment and the process characteristics numerical methods. The boundaries of the crater formed by the explosion, pressed and undisturbed areas of soil are determined. Numerical experiments were performed on the basis of the developed algorithm

Spatial Analogues of Numerical Quasiconformal Mapping Methods for Solving Problems of Bursts Parameters Identification

An approach to solving the problem of image reconstruction based on applied quasipotential tomographic data in the three-dimensional case is developed. It is based on the synthesis of spatial analogues of numerical quasiconformal mapping methods and algorithm for identifying the parameters of local bursts of homogeneous materials using similar methods on the plane. The peculiarity of the corresponding algorithm is taking into account (for each of the appropriate injections) the presence of only equipotential lines (with given values of the flow function or distributions of local velocities on them) and flow lines (with known potential distributions on them) at the domain boundary. Numerical experiments of simulative restoration of the environment structure are carried out.

Fredholm eigenvalues and quasiconformal geometry of polygons

An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory. This survey extends our previous survey of 2005 and presents the newapproaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

Quasiconformal Mapping Kernel Machine Learning-Based Intelligent Hyperspectral Data Classification for Internet Information Retrieval

Intelligent internet data mining is an important application of AIoT (Artificial Intelligence of Things), and it is necessary to construct large training samples with the data from the internet, including images, videos, and other information. Among them, a hyperspectral database is also necessary for image processing and machine learning. The internet environment provides abundant hyperspectral data resources, but the hyperspectral data have no class labels and no so high value for applications. So, it is important to label the class information for these hyperspectral data through machine learning-based classification. In this paper, we present a quasiconformal mapping kernel machine learning-based intelligent hyperspectral data classification algorithm for internet-based hyperspectral data retrieval. The contributions include three points: the quasiconformal mapping-based multiple kernel learning network framework is proposed for hyperspectral data classification, the Mahalanobis distance kernel function is as the network nodes with the higher discriminative ability than Euclidean distance-based kernel function learning, and the objective function of measuring the class discriminative ability is proposed to seek the optimal parameters of the quasiconformal mapping projection. Experiments show that the proposed scheme is effective for hyperspectral image classification and retrieval.

On nonhomeomorphic mappings with the inverse Poletsky inequality

As known, the modulus method is one of the most powerful research tools in the theory of mappings. Distortion of modulus has an important role in the study of conformal and quasiconformal mappings, mappings with bounded and finite distortion, mappings with finite length distortion, etc. In particular, an important fact is the lower distortion of the modulus under mappings. Such relations are called inverse Poletsky inequalities and are one of the main objects of our study. The use of these inequalities is fully justified by the fact that the inverse inequality of Poletsky is a direct (upper) inequality for the inverse mappings, if there exist. If the mapping has a bounded distortion, then the corresponding majorant in inverse Poletsky inequality is equal to the product of the maximum multiplicity of the mapping on its dilatation. For more general classes of mappings, a similar majorant is equal to the sum of the values of outer dilatations over all preimages of the fixed point. It the class of quasiconformal mappings there is no significance between the inverse and direct inequalities of Poletsky, since the upper distortion of the modulus implies the corresponding below distortion and vice versa. The situation significantly changes for mappings with unbounded characteristics, for which the corresponding fact does not hold. The most important case investigated in this paper refers to the situation when the mappings have an unbounded dilatation. The article investigates the local and boundary behavior of mappings with branching that satisfy the inverse inequality of Poletsky with some integrable majorant. It is proved that mappings of this type are logarithmically Holder continuous at each inner point of the domain. Note that the Holder continuity is slightly weaker than the classical Holder continuity, which holds for quasiconformal mappings. Simple examples show that mappings of finite distortion are not Lipschitz continuous even under bounded dilatation. Another subject of research of the article is boundary behavior of mappings. In particular, a continuous extension of the mappings with the inverse Poletsky inequality is obtained. In addition, we obtained the conditions under which the families of these mappings are equicontinuous inside and at the boundary of the domain. Several cases are considered: when the preimage of a fixed continuum under mappings is separated from the boundary, and when the mappings satisfy normalization conditions. The text contains a significant number of examples that demonstrate the novelty and content of the results. In particular, examples of mappings with branching that satisfy the inverse Poletsky inequality, have unbounded characteristics, and for which the statements of the basic theorems are satisfied, are given.

Research in the Field of Geometric Analysis at Volgograd State University

This article discusses the main directions of research in geometric analysis, which were conducted and are being carried out by the scientific mathematical school of Volgograd State University. The results of the founder of the scientific school, Doctor of Physics and Mathematics, Professor Vladimir Mikhailovich Miklyukov and his students are summarized. These results concern the solution of a number of problems in the field of quasiconformal flat mappings and mappings with bounded distortion of surfaces and Riemannian manifolds, the theory of minimal surfaces and surfaces of prescribed mean curvature, surfaces of zero mean curvature in Lorentz spaces, as well as problems associated with the study of the stability of such surfaces. In addition, the results of the study of various classes of triangulations — an object that appears at the junction of research in the field of geometric analysis and computational mathematics — are noted. Besides, this review discusses papers that use the Fourier decomposition method for solutions of the Laplace — Beltrami equations and the stationary Schr¨odinger equation with respect to the eigenfunctions of the corresponding boundary value problems. In particular, the authors give the results on finding capacitive characteristics that allowed for the first time to formulate and prove the criteria for the fulfillment of various theorems of Liouville type and the solvability of boundary value problems on model and quasimodel Riemannian manifolds. The role of the equivalent function method is also indicated in the study of such problems on manifolds of a fairly general form. In addition to this, the article gives an overview of the results concerning estimates of calculating error integral functionals and convergence of piecewise polynomial solutions of nonlinear variational type equations: minimal surface equations, equilibrium equations capillary surface and equations of biharmonic functions.

Sharp Rellich–Sobolev inequalities and weighted Adams inequalities involving Hardy terms for bi-Laplacian

Motivated by a recent work of Dong, Lam and Lu concerning the sharp weighted Trudinger–Moser and Caffarelli–Kohn–Nirenberg inequalities, we derive a sharp Rellich–Sobolev inequality on Rn with n≥5 and weighted Adams inequalities involving Hardy terms on R4. The procedure is based on a quasi-conformal mapping type transform and decomposition into spherical harmonics since the symmetrization arguments do not work in dealing with these inequalities.