arXiv subjects

Vladimir Ryazanov

Publications and source records attributed to Vladimir Ryazanov.

At least 19 recordsLinked to original sources

arXiv2026

A proof of the Riemann hypothesis on zeros of $ζ-$function

In his famous presentation at the International Congress of Mathematicians held in Paris in 1900, David Hilbert included the Riemann Hypothesis on zeros of $ζ-$function as number 8 in his list of 23 challenging problems published later. After over 150 years, it is one of the few on that list that have not been solved. At present many mathematicians consider it the most important unsolved problem in mathematics. Recall that, exactly one hundred years later, the Clay Mathematics Institute has published a list of 7 unsolved problems for the 21st century, including 6 unresolved problems from the Hilbert list, offering a reward of one million dollars for a solution to any of these problems. One of them is the {\bf Riemann hypothesis}, i.e. a conjecture that the so-called Riemann zeta function has as its zeros only complex numbers with real part $1/2$ in addition to its trivial zeros at the negative even integers. It was proposed by Bernhard Riemann in his 1859 paper. The Riemann zeta function plays a great role in analytic number theory as well as in physics, probability theory and applied statistics. In this preprint, applying the known Beurling--Nyman criterion, it is just proved the Riemann hypothesis.

math.GM↗

arXiv2025

Modulus estimates and cavitation in higher dimensions

We explore the phenomenon of cavitation in higher-dimensional elasticity, defining it as the mapping of a punctured ball onto a non-degenerate ring domain. Crucially, for the class of locally quasiconformal mappings (or more general mappings) defined on the punctured ball $0<|x|<1$ in $\mathbb R^n$ that we examine, cavitation is equivalent to a failure of continuous extension to the origin. While existing modulus estimates prove insufficient for reliably detecting cavitation in this setting, our study establishes refined modulus bounds. This is achieved by introducing a novel directional dilatation which, in conjunction with the known angular dilatation, overcomes the limitations of previous methods. We illustrate our theoretical findings with several examples that demonstrate both cavitation occurrence and its absence.

math.CV↗

arXiv2022

Hilbert and Poincare problems for semi-linear equations in domains with rectifiable boundaries

In the last paper \cite{R7}, it was studied Hilbert, Poincare and Neumann boundary-value problems with arbitrary measurable data for generalized analytic functions and generalized harmonic functions with applications to the relevant problems of mathematical physics. The present paper is devoted to the study of the boundary-value problems with arbitrary measurable boundary data in domains with rectifiable boundaries for the corresponding semi-linear equations with suitable nonlinear sources. For this purpose, here it is constructed completely continuous operators generating nonclassical solutions of the Hilbert and Poincare boundary-value problems with arbitrary measurable data for the Vekua type equations and the Poisson equations, respectively. On this base, it is first proved the existence of solutions of the Hilbert boundary-value problem with arbitrary measurable data in any domains with rectifiable boundaries for the nonlinear equations of the Vekua type. It is necessary to note that our approach is based on the geometric interpretation of boundary values as angular (along nontangential path) limits in comparison with the classical variational approach in PDE. The latter makes it is also possible to obtain the theorem on the existence of nonclassical solutions of the Poincare boundary-value problem on the directional derivatives and, in particular, of the Neumann problem with arbitrary measurable data to the Poisson equations with nonlinear sources in Jordan domains with rectifiable boundaries. As consequences, then it is given a series of applications of this theorem to some problems of mathematical physics describing, for instance, such phenomena as physical and chemical absorption with diffusion, plasma states, stationary burning etc.

math.CV↗

arXiv2020

Neumann and Poincare problems for Poisson's equations with measurable data

The research of the Dirichlet problem with arbitrary measurable datafor harmonic functions is due to the famous dissertation of Luzin. The present paper is devoted to various theorems on the existence of nonclassical solutions of the Hilbert and Riemann boundary value problems with arbitrary measurable data for generalized analytic functions by Vekua and the corresponding applications to the Neumann and Poincare problems for generalized harmonic functions. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. Here it is proved the existence theorems on solutions of the Hilbert boundary value problem with arbitrary measurable data for generalized analytic functions in arbitrary Jordan domains with rectifiable boundaries in terms of the na\-tu\-ral parameter and angular (nontangential) limits, moreover, to arbitrary Jordan domains in terms of harmonic measure and principal asymptoticvalues. Moreover, it is established the existence theorems on solutions for the appropriate boun\-da\-ry value problems of Hilbert and Riemann with arbitrary measurable data along the Bagemihl--Seidel systems of Jordan arcs terminating at the boundary in arbitrary domains whose boundaries consist of finite collections of rectifiable Jordan curves. On this basis, it is established the corresponding existence theorems for the Poincare boundary value problem on the directional derivatives and, in particular, for the Neumann problem with arbitrary measurable data to the Poisson equations.

math.CV↗

arXiv2019

The Stieltjes integrals in the theory of harmonic and analytic functions

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 singular Hilbert-Stieltjes integral in the sense of the principal value. These results hold for arbitrary $2π-$periodic bounded integrands that are differentiable a.e. and, in particular, for integrands of the class ${\cal {CBV}}$ (countably bounded variation).

math.CV↗

arXiv2019

The Dirichlet problem for semi-linear equations

We study the Dirichlet problem for the semi--linear partial differential equations ${\rm div}\,(A\nabla u)=f(u)$ in simply connected domains $D$ of the complex plane $\mathbb C$ with continuous boundary data. We prove the existence of the weak solutions $u$ in the class $C\cap W^{1,2}_{\rm loc}(D)$ if a Jordan domain $D$ satisfies the quasihyperbolic boundary condition by Gehring--Martio. An example of such a domain that fails to satisfy the standard (A)--condition by Ladyzhenskaya--Ural'tseva and the known outer cone condition is given. We also extend our results to simply connected non-Jordan domains formulated in terms of the prime ends by Caratheodory. Our approach is based on the theory of the logarithmic potential, singular integrals, the Leray--Schauder technique and a factorization theorem in \cite{GNR2017}. This theorem allows us to represent $u$ in the form $u=U\circω,$ where $ω(z)$ stands for a quasiconformal mapping of $D$ onto the unit disk ${\mathbb D}$, generated by the measurable matrix function $A(z),$ and $U$ is a solution of the corresponding quasilinear Poisson equation in the unit disk ${\mathbb D}$. In the end, we give some applications of these results to various processes of diffusion and absorption in anisotropic and inhomogeneous media.

math.CV↗

arXiv2018

Prime ends and mappings on Riemann surfaces

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.

math.CV↗

arXiv2018

Correlation of boundary behavior of conjugate harmonic functions

It is established 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 $E$. The result is extended to arbitrary Jordan domains with rectifiable boundaries in terms of angular limits and of the natural parameter.

math.CV↗

arXiv2017

Toward the theory of semi-linear equations

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 equation in a canonical domain and a quasiconformal mapping agreed with a matrix-valued measurable coefficient appearing in the divergence part of the equation. The latter makes it possible, in particular, to remove the regularity restrictions on the boundary in the study of boundary value problems for such semilinear equations.

math.CV↗

arXiv2017

On the Riemann-Hilbert problem for the Beltrami equation

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 with respect to the absolute harmonic measure (logarithmic capacity). Moreover, it is shown that the dimension of the spaces of the given solutions is infinite. One more section was added for the case of coefficients of countably bounded variation.

math.CV↗

arXiv2016

On Sobolev's mappings on 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

math.CV↗

arXiv2016

On Neumann and Poincare problems for Laplace equation

It is proved the existence of nonclassical 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 found solutions have the infinite dimension.

math.CV↗

arXiv2015

On Hilbert, Riemann, Neumann and Poincare problems for plane quasiregular mappings

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 spaces of the found solutions have the infinite dimension for prescribed families of Jordan arcs terminating in almost every boundary point. We give also applications of results obtained by us for the Beltrami equations to the boundary value problems of Dirichlet, Riemann, Neumann and Poincare for A-harmonic functions in the plane.

math.CV↗

arXiv2015

On Hilbert and Riemann problems. An alternative approach

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 shown that the spaces of the found solutions have the infinite dimension for prescribed collections of Jordan arcs terminating in almost every boundary point. Similar results are proved for the Riemann problem.

math.CV↗

arXiv2015

On multivalent solutions of Riemann-Hilbert problem

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 corresponding reinforced criterion for domains with rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Furthemore, it is shown that the dimension of the spaces of these solutions is infinite.

math.CV↗

arXiv2015

Ring homeomorphisms and prime ends

We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ of a Beltrami equation $\overline{\partial}f=μ\,\partial f$ in a domain $D\subseteq\Bbb C$ is the so--called ring $Q-$homeomorphism with $Q(z)=K^T_μ(z, z_0)$ where $K^T_μ(z, z_0)$ is the tangent (angular) dilatation quotient of the equation with respect to an arbitrary point $z_0\in {\overline{D}}$. In this connection, we develop the theory of the boundary behavior of the ring $Q-$homeomorphisms with respect to prime ends. On this basis, we show that, for wide classes of degenerate Beltrami equations $\overline{\partial}f=μ\,\partial f$, there exist regular solutions of the Dirichlet problem in arbitrary simply connected domains in $\Bbb C$ and pseudoregular and multivalent solutions in arbitrary finitely connected domains in $\Bbb C$ with boundary datum $φ$ that are continuous with respect to the topology of prime ends.

math.CV↗

arXiv2015

The Dirichlet problem and prime ends

It is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ${\bar{\partial}}f=μ\,{\partial}f$ 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 $μ$, it is proved the existence of regular solutions of its Dirichlet problem in arbitrary simply connected domains and pseudoregular as well as multivalent solutions in arbitrary finitely connected domains with continuous boundary data in terms of prime ends.

math.CV↗

arXiv2015

The theory of prime ends and spatial mappings

It is given a canonical representation of prime ends in regular spatial domains and, on this basis, it is studied the boundary behavior of the so-called lower Q-homeomorphisms that are the natural generalization of the quasiconformal mappings. In particular, it is found a series of effective conditions on the function Q(x) for a homeomorphic extension of the given mappings to the boundary by prime ends in domains with regular boundaries. The developed theory is applied, in particular, to mappings of the classes of Sobolev and Orlicz-Sobolev and also to finitely bi-Lipschitz mappings that a far-reaching extension of the well--known classes of isometric and quasiisometric mappings.

math.CV↗