arXiv subjects

E. Yakubov

Publications and source records attributed to E. Yakubov.

8 recordsLinked to original sources

arXiv2023

On Dirichlet problem for degenerate Beltrami equations with sources

The present paper is devoted to the study of the Dirichlet problem ${\rm{Re}}\,ω(z)\toφ(ζ)$ as $z\toζ,$ $z\in D,ζ\in \partial D,$ with continuous boundary data $φ:\partial D\to\mathbb R$ for Beltrami equations $ω_{\bar{z}}=μ(z) ω_z+σ(z)$, $|μ(z)|<1$ a.e., with sources $σ:D\to\mathbb C$ in the case of locally uniform ellipticity. In this case, we establish a series of effective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on singularities of the equations at the boundary for existence, representation and regularity of solutions in arbitrary bounded domains $D$ of the complex plane $\mathbb C$ with no boun\-da\-ry component degenerated to a single point for sources $σ$ in $L_p(D)$, $p>2$, with compact support in $D$. Moreover, we prove in such domains existence, representation and regularity of weak solutions of the Dirichlet problem for the Poisson type equation ${\rm div} [A(z)\nabla\,u(z)] = g(z)$ whose source $g\in L_p(D)$, $p>1$, has compact support in $D$ and whose mat\-rix valued coefficient $A(z)$ guarantees its locally uniform ellipticity.

math.CV↗

arXiv2022

To the theory of semi-linear Beltrami equations

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 prove existence of regular solutions of the semi-linear Beltrami equations with no boundary conditions. Moreover, here we derive their representation through solutions of the Vekua type equations and generalized analytic functions with sources. As consequences, it is given a series of applications of these results to semi-linear Poisson type equations and to the corresponding equations of mathematical physics describing such phenomena as diffusion with physical and chemical absorption, plasma states and stationary burning in anisotropic and inhomogeneous media. The second part of the paper contains existence, representation and regularity results for nonclassical solutions to the Hilbert (Dirichlet) boundary value problem for semi-linear Beltrami equations and to the Poincare (Neumann) boundary value problem for semi-linear Poisson type equations with arbitrary boundary data that are measurable with respect to logarithmic capacity.

math.CV↗

arXiv2022

On boundary value problems for inhomogeneous Beltrami equations

We give here results on the existence of nonclassical solutions of the Hilbert boundary value problem in terms of the so-called angular limits (along nontangent curves to the boundary) for Beltrami equations with sources in Jordan domains satisfying the quasihyperbolic boundary condition by Gehring-Martio, generally speaking, without $(A)-$condition by Ladyzhenskaya-Ural'tseva and, in particular, without the known outer cone condition that were standard for boundary value problems in the PDE theory. Assuming that the coefficients of the problem are functions of countable bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of locally Hölder continuous solutions of the problem. Moreover, we prove similar results on the Hilbert boundary value problem with its arbitrary measurable boundary data as well as coefficients in finitely connected Jordan domains along the so-called general Bagemihl-Seidel systems of Jordan arcs to their boundaries. In the same terms, we formulate theorems on the existence of regular solutions of the known Riemann boundary value problems, including nonlinear, with arbitrary measurable coefficients for the nonhomogeneous Beltrami equations. We give also the representation of obtained solutions through the so-called generalized analytic functions with sources. Finally, we formulate similar results on Poincare and Neumann problems for the Poisson type equations that are main in hydromechanics (mechanics of incompressible fluids) in anisotropic and inhomogeneous media. We give here the representation of their solutions through the so-called generalized harmonic functions with sources that describe the corresponding physical processes in iso\-tro\-pic and homogeneous media.

math.AP↗

arXiv2022

Beltrami equations and mappings with asymptotic homogeneity at infinity

First of all, we prove that the BMO condition by John-Nirenberg leads in the natural way to the asymptotic homogeneity at the origin of regular homeomorphic solutions of the degenerate Beltrami equations. Then on this basis we establish a series of criteria for the existence of regular homeomorphic solutions of the degenerate Beltrami equations in the whole complex plane with asymptotic homogeneity at infinity. These results can be applied to the fluid mechanics in strictly anisotropic and inhomogeneous media because the Beltrami equation is a complex form of the main equation of hydromechanics.

math.CV↗

arXiv2022

Toward the theory of Dirichlet problem for the degenerate Beltrami equations

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 $\overline{\partial}f=μ\,\partial f$ in $\mathbb C$ whose complex coefficients $μ$ have compact supports. On this basis, we establish criteria for existence and representation of regular discrete open solutions for the Dirichlet problem with continuous data to degenerate Beltrami equations in arbitrary simply connected bounded domains $D$ in $\mathbb C$. Moreover, we obtain similar criteria for the existence of multi-valued solutions $f$ in the spirit of the theory of multi-valued analytic functions in arbitrary bounded domains $D$ in $\mathbb C$ with no boundary component degenerated to a single point. Note that the latter request is necessary and that the real parts $u$ of such solutions $f$ are the so-called $A-$harmonic functions, i.e., single-valued continuous weak solutions of elliptice quations ${\rm div}\, (A\cdot\nabla u)=0$ with matrix-valued coefficients $A$ associated with $μ$. Thus, the results can be applied to potential theory in anisotropic and inhomogeneous media.

math.CV↗

arXiv2019

On Hilbert problem for Beltrami equations in quasihyperbolic domains

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 measurable with respect to logarithmic capacity. It is shown that the dimension of the spaces of such solutions is infinite. It is also given applications to the Dirichlet, Neumann and Poincare boundary value problems for analogs of the Laplace equation in anisotropic and inhomogeneous media.

math.CV↗

arXiv2010

Integral conditions in the theory of the Beltrami equations

It is shown that many recent and new results on the existence of ACL homeomorphic (and more strong) solutions for the Beltrami equations with integral constraints follow from our extension of the well--known Lehto existence theorem.

math.CV↗

arXiv2010

On integral conditions in the mapping theory

It is established interconnections between various integral conditions that play an important role in the theory of space mappings and in the theory of degenerate Beltrami equations in the plane.

math.CV↗