Vladimir Illich Ryazanov

institute of Applied Mathematics and Mechnics of National Academy of Sciences of Ukraine, Slavyansk, Ukraine · Head of the Function Theory Department

The present paper is devoted to the study of semi-linear Beltrami equations which are closely relevant to the corresponding semi-linear Poisson type equations of mathematical physics on the plane in anisotropic and inhomogeneous media. In its first part, applying completely continuous ope\-ra\-tors by Ahlfors-Bers and Leray--Schauder approach, we p...

We establish some criteria for the existence of regular homeomorphic solutions of the degenerate Beltrami equations in the complex plane with asymptotic homogeneity at infinity.

In this article, first we give a general lemma on the existence of regular homeomorphic solutions $f$ with the hydrodynamic normalization $f(z)=z+o(1)$ as $z\to\infty$ to the degenerate Beltrami equations $\overline{\partial}f=\mu\,\partial f$ in $\mathbb C$ whose complex coefficients $\mu$ have compact supports. On this basis, we establish criteri...

The study of the Dirichlet problem in the unit disk 𝔻 with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua [48] has been devoted to boundary-value problems (only with Hölder continuous data) for the generalized analytic functions, i.e., continuous complex value...

We study various Stieltjes integrals as Poisson-Stieltjes, conjugate Poisson-Stieltjes, Schwartz-Stieltjes and Cauchy-Stieltjes and prove theorems on the existence of their finite angular limits a.e. in terms of the Hilbert-Stieltjes integral. These results hold for arbitrary $2\pi -$periodic bounded Borel integrands that are differentiable a.e. an...

It is known that if a harmonic function u on the unit disk \({\mathbb {D}}\) in \({\mathbb {C}}\) has angular limits on a measurable set E of the unit circle \(\partial {\mathbb {D}}\), then its conjugate harmonic function v in \({\mathbb {D}}\) also has angular limits a.e. on E and both boundary functions are finite a.e. and measurable on E. This...

We study the Dirichlet problem with continuous boundary data in simply connected domains D of the complex plane for the semi-linear partial differential equations whose linear part has the divergent form. We prove that if a Jordan domain D satisfies the so-called quasihyperbolic boundary condition, then the problem has regular (continuous) weak sol...

It is studied the Hilbert boundary value problem for the nondegenerate Beltrami equations in domains $D$ of the complex plane $\mathbb C$ with the so--called quasihyperbolic boundary condition. It is proved the existence of solutions of this problem with coefficients of countably bounded variation and with arbitrary boundary data that are measurabl...

We study the Dirichlet problem for the semi--linear partial differential equations in the simple connected domains $D$ in $\mathbb C$, the linear part of which is written in a divergence (anisotropic !) form. Thanking to a factorization theorem established by us earlier in \cite{GNR2017}, the problem is reduced to the Dirichlet problem for the corr...

Assume that Ω is a domain in the complex plane ℂ and A(z) is a symmetric 2×2 matrix function with measurable entries, detA = 1; and such that 1/K|ξ|2 ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|2, ξ ∈ ℝ2, 1 ≤ K < ∞ . In particular, for semilinear elliptic equations of the form div (A(z)∇u(z)) = f(u(z)) in Ω; we prove a factorization theorem that asserts that every weak sol...

The criteria for continuous and homeomorphic extensions to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends by Carath´eodory are proved.

In this paper we study the semilinear partial differential equations in the plane the linear part of which is written in a divergence form. The main result is given as a factorization theorem. This theorem states that every weak solution of such an equation can be represented as a composition of a weak solution of the corresponding isotropic equati...

It is proved the existence of generalized solutions of the Neumann problem for the harmonic functions in the Jordan rectifiable domains with arbitrary measurable boundary distributions of normal derivatives. The same is stated for the partial case of the Poincare problem on directional derivatives. Moreover, it is shown that the spaces of the given...

We prove criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.

It is proved criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.

The survey is devoted to recent advances in nonclassical solutions of the main boundary-value problems such as the well-known Dirichlet, Hilbert, Neumann, Poincaré, and Riemann problems in the plane. Such solutions are essentially different from the variational solutions of the classical mathematical physics and based on the nonstandard point of vi...

Assume that Ω is a regular domain in the complex plane ℂ, and A(z) is a symmetric 2×2 matrix with measurable entries, det A = 1, and such that 1/K|ξ|2 ≤ 〈A(z)ξ, ξ〉 ≤ K|ξ|2, ξ ∈ ℝ2, 1 ≤ K < ∞. We study the blow-up problem for a model semilinear equation div (A(z)∇u) = eu in Ω and show that the well-known Liouville–Bieberbach function solves the prob...

In the present paper, it is studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The devel...

We establish a series of new criteria of equicontinuity and, hence, normality of the mappings of Orlicz–Sobolev classes in terms of inner dilatations.

A canonical representation of prime ends is obtained in the case of regular spatial domains, and the boundary behavior is studied for the so-called lower Q-homeomorphisms, which generalize the quasiconformal mappings in a natural way. In particular, a series of efficient conditions on a function Q are found for continuous and homeomorphic extendibi...

In terms of dilatations, it is proved a series of criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between regular domains on the Riemann surfaces

In terms of dilatations, it is proved a series of criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between regular domains on the Riemann surfaces

Generalized solvability of the classical boundary-value problems for analytic and quasiconformal functions in arbitrary Jordan domains with boundary data that are measurable with respect to the logarithmic capacity is established. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. Finally, some applications to...

We proved the existence of multivalent solutions with the infinite number of branches for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic...

We prove analogs of theorems of Lusin and Gehring in terms of logarithmic capacity. On this basis, for the nondegenerate Beltrami equations in the quasidisks and, in particular, in smooth (and Lipschitz) Jordan domains, we establish the existence of regular solutions of the Riemann-Hilbert problem with coefficients of bounded variation and boundary...

For the nondegenerate Beltrami equations in the quasidisks and, in particular, in smooth Jordan domains, we prove the existence of regular solutions of the Riemann–Hilbert problem with coefficients of bounded variation and boundary data that are measurable with respect to the absolute harmonic measure (logarithmic capacity).

Recall that the Hilbert (Riemann-Hilbert) boundary value problem was recently solved in \cite{R1} for arbitrary measurable coefficients and for arbitrary measurable boundary data in terms of nontangential limits and principal asymptotic values. Here it is developed a new approach making possible to obtain new results on tangential limits. It is sho...

We present a canonical representation of prime ends in regular domains and, on this basis, study the boundary behavior of the so-called lower Q-homeomorphisms obtained as a natural generalization of quasiconformal mappings. We establish a series of effective conditions imposed on a function Q(x) for the homeomorphic extension of given mappings with...

Recall that the Hilbert (Riemann-Hilbert) boundary value problem for the Beltrami equations was recently solved for general settings in terms of nontangential limits and principal asymptotic values. Here it is developed a new approach making possible to obtain new results on tangential limits in multiply connected domains. It is shown that the spac...

Recall that the Hilbert (Riemann-Hilbert) boundary value problem was recently solved for arbitrary measurable coefficients and for arbitrary measurable boundary data in terms of nontangential limits and principal asymptotic values. Here it is developed a new approach making possible to obtain new results on tangential limits. It is shown that the s...

It is proved the existence of single-valued analytic solutions in the unit disk and multivalent analytic solutions in domains bounded by a finite collection of circles for the Riemann-Hilbert problem with coefficients of sigma-finite variation and with boundary data that are measurable with respect to logarithmic capacity. It is shown that these sp...

We first study the boundary behavior of ring Q-homeomorphisms in terms of Carathéodory’s prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation \( \overline{\partial} \) f = μ∂f in arbitrary bounded finitely connected domains D of the complex plane ℂ.

Generalized solvability of the classical boundary value problems for analytic and quasiconformal functions in arbitrary Jordan domains with boundary data that are measurable with respect to the logarithmic capacity is established. Moreover, it is shown that the spaces of the found solutions have the infinite dimension. Finally, some applications to...

It is proved the existence of multivalent solutions for the Riemann-Hilbert problem in the general settings of finitely connected domains bounded by mutually disjoint Jordan curves, measurable coefficients and measurable boundary data. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corre...

It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.

It is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ${\overline{\partial}}f=\mu\,{\partial}f$ of the Sobolev class $W^{1,1}_{\rm loc}$ with respect to prime ends of domains. On this basis, under certain conditions on the complex coefficient ${\mu}$, it is proved the existence of regular solutions...

We show that every homeomorphic solution of a Beltrami equation with generalized derivatives in a domain D ⊆ C is the so–called ring Q−homeomorphism with Q(z) is equal to the tangent (angular) dilatation quotient of the equation with respect to an arbitrary point in the closure of D. In this connection, we develop the theory of the boundary behavio...

We first study the boundary behavior of ring Q-homeo-morphisms in terms of Carathéodory's prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation ∂f = µ ∂f in arbitrary bounded finitely connected domains D of the complex plane C.

In terms of prime ends by Caratheodory, it is studied the boundary behavior of the socalled lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prim...

It is shown that, under a Calderón type condition on the function ϕ, the continuous open mappings that belong to the Orlicz-Sobolev classes W1,ϕloc have total differential almost everywhere; this generalizes the well-known theorems of Gehring- Lehto-Menchoff in the case of R2 and of Väisälä in Rn, n ≥ 3. Appropriate examples show that the Calderón...

We show that homeomorphisms $f$ in ${\Bbb R}^n$, $n\geqslant3$, of finite distortion in the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with a condition on $\varphi$ of the Calderon type and, in particular, in the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$ are the so-called lower $Q$-homeomorphisms, $Q(x)=K^{\frac{1}{n-1}}_I(x,f)$, where...

We show that homeomorphic solutions of the Beltrami equations satisfy certain modular inequalities. On this basis, we develop the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichlet problem to the Beltrami equation in Jordan domains...

It is proved the existence of solutions for the Riemann-Hilbert problem in the fairly general settings of arbitrary Jordan domains, mea-surable coefficients and measurable boundary dates. The theorem is for-mulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with...

Under a condition of the Calderon type, we show that a homeomorphism of fi�nite distortion in the Orlicz-Sobolev classes is the so called lower Q-homeomorphism where Q is equal to its inner dilatation in the degree 1/(n-1) and the so called ring Q-homeomorphism where Q is equal to its inner dilatation. Similar statements are valid also for the so c...

We show that arbitrary homeomorphic solutions to the Beltrami equations with generalized derivatives satisfy certain moduli inequalities. On this basis, we develope the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichlet problem to...

We give necessary and sufficient conditions of equicontinuity of so-called ring Q-mappings that will have significant applications to the general Beltrami equation, a complex form of one of the main equations of the mathematical physics in the plane, as well as to its many-dimensional analogs and to the Orlicz–Sobolev mappings in space.

We show that homeomorphic W loc 1,1 solutions of the Beltrami equations \(\overline \partial f = \mu \partial f\) satisfy certain modular inequalities. On this basis, we develop the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichle...

It is proved that homeomorphisms of the Orlicz-Sobolev class W loc 1, φ can be continuously extended to the boundaries of some domains if the function φ defining this class satisfies a Carderón-type condition and the outer dilatation K f of the mapping f satisfies the divergence condition for integrals of special form. In particular, the result hol...

It is proved that the dimension of the space of solutions of the Dirichlet problem for the harmonic functions with nontangential limits in the unit disk is infinite.

For the nondegenerate Beltrami equations in the quasidisks and, in particular, in smooth Jordan domains, it is proved the existence of regular solutions of the Riemann-Hilbert problem with coefficients of bounded variation and boundary data that are measurable with respect to the absolute harmonic measure (logarithmic capacity).

It is developed the theory of the Dirichlet problem for harmonic functions. On this basis, for the nondegenerate Beltrami equations in the quasidisks and, in particular, in the smooth domains, it is proved the existence of regular solutions of the Riemann-Hilbert problem with coefficients of bounded variation and boundary data that are measurable w...

It is shown that homeomorphisms f in \( {{\mathbb{R}}^n} \), n ≥ 2, with finite Iwaniec distortion of the Orlicz–Sobolev classes W 1,φloc under the Calderon condition on the function φ and, in particular, the Sobolev classes W 1,φloc, p > n - 1, are differentiable almost everywhere and have the Luzin (N) -property on almost all hyperplanes. This en...

We prove a series of criteria for homeomorphic extension of spatial mappings to the boundary by prime ends

We study the Dirichlet problem for general degenerate Beltrami equation ${\overline {\partial}}f\, =\, \mu {\partial f}+\nu {\overline {\partial f}}$ in the unit disk D in C. Given an arbitrary analytic function A with isolated singularities in D, we find criteria for the existence of a solution f of the form f = A◦ω where ω stands for a regular...

We show the existence of regular solutions of the Dirichlet problem with continuous boundary data for the wide classes of the Beltrami equations in arbitrary Jordan domains of the complex plane

The book is devoted to an exposition of results of investigations carried out mainly over the last 15--20 years concerning the theory of local topological and geometric properties of mappings in the Euclidean spaces obtained jointly with Professors Olli Martio, Matti Vuorinen (Finland), Christopher Bishop (USA) and Anatoly Golberg (Israel). Quasi...

Various theorems on convergence of general spatial homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring Q-homeomorphisms are obtained. In particular, it is established that a family of all ring Q-homeomorphisms f in ℝn fixing two points is compact provided that the function Q is of...

It is proved the existence of a solution for the Riemann-Hilbert problem in the fairly general setting of arbitrary Jordan domains, coefficients of bounded variation and measurable dates. The theorem is formulated in terms of harmonic measure and principal asymptotic values. It is also given the corresponding reinforced criterion for domains with a...

This book is devoted to the recent advances in the theory of local topological and geometric properties of mappings in the n - dimensional Euclidean space. Quasiconformal, quasiregular and bi-Lipschitz mappings and the methods to study local behavior of these mappings without differentiability assumptions form the core of the book. The book consist...

The purpose of this book is to present recent advances in the theory of local properties of mappings in the plane. The main emphasis is in the almost ev- erywhere di®erentiable homeomorphic mappings. Thus quasiconformal and bi- Lipschitz mappings and the methods to study these mappings form the core of the book. Quasiconformal mappings have turned...

We study the Dirichlet problem for the general degenerate Beltrami equations $ \bar{\partial}f=\mu \partial f+\nu \overline{{\partial f}} $ in a Jordan domain. Some criteria for the existence of regular solutions are given.

Various theorems on the convergence of general spatial homeomorphisms are proved and, on this basis, convergence theorems for classes of the so-called ring Q--homeomorphisms are obtained. These results will have wide applications to Sobolev's mappings.

We show that every homeomorphic generalized solution f to a Beltrami equation in a domain D ⊆ C is the so–called lower Q−homeomorphism with Q which is equal to the tangent dilatation of f and develop the theory of the boundary behavior of such solutions. Then, on this basis, we show that, for wide classes of degenerate Beltrami equations, there exi...

Criteria on the existence of regular solutions of the Dirichlet problem for the degenerate Beltrami equation ∂ ¯f=ν∂f ¯ in a Jordan domain of the complex plane ℂ are given.

We give a brief overview of our results in the theory of variations for classes of regular solutions to the degenerate Beltrami equation with constraints of set-theoretic and integral type for the coefficients. The variational maximum principle and other necessary extremal conditions are formulated and applications to one of the main equations of m...

We establish a series of criteria for the existence of regular solutions of the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. We also formulate the corresponding criteria for the existence of pseudoregular and multivalued solutions of the Dirichlet problem in the case of finitely connected domains.

We establish a series of criteria on the existence of regular solutions for the Dirichlet problem to general degenerate Beltrami equations ${\bar{\partial}}f = \mu {\partial f}+\nu {\bar{\partial f}}$ in arbitrary Jordan domains in $\C$.

We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ to a Beltrami equation $\bar{\partial}f=\mu \partial f$ in a domain $D\subset\Bbb C$ is the so--called lower $Q-$homeomorphism with $Q(z)=K^T_{\mu}(z, z_0)$ where $K^T_{\mu}(z, z_0)$ is the tangent dilatation of $f$ with respect to an arbitrary point $z_0\in {\bar{D}}$ and develop the...

First of all, we establish compactness of continuous mappings of the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$ in ${\Bbb R}^n\,,$ $n\ge 3\,,$ with one fixed point. Then we give a series of theorems on convergence of the O...

Various theorems on convergence of general space homeomorphisms are proved and, on this basis, theorems on convergence and compactness for classes of the so-called ring $Q$--homeomorphisms are obtained. In particular, it was established by us that a family of all ring $Q$--homeomorphisms $f$ in ${\Bbb R}^n$ fixing two points is compact provided tha...

The book is a summation of many years’ work in the study of general Beltrami equations with singularities. This is not only a summary of our own long– term collaboration but also that of many other authors in the field.We show that our geometric approach based on the modulus and capacity developed by us makes it possible to derive the main known ex...

We have constructed variations for classes of regular solutions of the degenerate Beltrami equation with constraints of the set-theoretic and integral types for the complex coefficient and, on this basis, proved the variational principles of maximum and other necessary conditions of extremum. Some applications to equations of mathematical physics a...

In this chapter, the BMO-quasiconformal and BMO-quasiregular mappings in the plane are studied. This includes distortion, existence, uniqueness, representation, integrability, convergence and removability theorems, the reflection principle, boundary behavior, and mapping properties.

Recall that in our notation, a µ-homeomorphism in a domain D,D ⊂ ℂ is an ACL homeomorphic solution of (B) in D; see Sect. 1.5. For some functions µ with |µ(z)| ≤ 1 a.e. and ||µ||∞ = 1, there are no µ-homeomorphisms, i.e., homeomorphic solutions of (B), as illustrated below in Sects. 4.1.1 and 4.1.2. Even when a µ-homeomorphism exists, it is not kno...

In this chapter, we introduce and study plane ring Q-homeomorphisms. This study is then applied to deriving general principles on the existence of strong ring solutions to the Beltrami equation extending and strengthening earlier results; see the next chapter.

Roughly speaking, this term refers to equation (B) in the case where μ is a measurable complex-valued function in a domain D in ℂ with |μ| < 1 a.e. in parts of D and |μ| > 1 a.e. in other parts of D. With further assumptions on μ, every nonconstant solution f : D → ℂ of (B), i.e., an ACL mapping which satisfies (B) a.e., is locally quasiregular (an...

The Beltrami equations of the first type $$f_{\bar{z}} = \mu(z)f_{z}$$ (9.1.1) is the basic equation for the theory of quasiconformalmapping in the complex plane; see, e.g., [9,30,44] and [152]. The well-known measurable mapping theorem solves the problem on the existence and uniqueness for the classical case.

Consider the Beltrami equation (B) in Sect. 1.1 with ||µ||∞<1, and let f : D → ℂ be its homeomorphic solution. Since |µ| < 1 a.e., f is sense preserving, and since ||µ||∞<1, f is a quasiconformal mapping.

The class BMO was introduced by John and Nirenberg in the paper [122] and soon became an important concept in harmonic analysis, partial differential equations, and related areas; see, e.g., [21, 24, 84, 103, 200] and [239].

We study the problems of the continuous and homeomorphic extension to the boundary of lower Q-homeomorphisms between domains on Riemannian manifolds and formulate the corresponding consequences for homeomorphisms with finite distortion in the Orlicz–Sobolev classes $ W_{loc}^{1,\varphi } $ under a condition of the Calderon type for the function φ...

A su-cient condition for the existence of a local folding solution to an alternating Beltrami equation is presented, and the uniformization of local folding solutions is studied.

В пятом томе серии «Задачи и методы: математика, механика, кибернетика» представлены исследования по теории конформных и квазиконформных отображений и их обобщений. Первая часть монографии посвящена геометрической теории аналитических функций и содержит решение ряда трудных экстремальных задач этой теории. Вторая часть связана с исследованием локал...