On the existence of solutions of the Beltrami equations with conditions on inverse dilatations

We have found one of possible conditions under which the degenerate Beltrami equation has a continuous solution of the Sobolev class. This solution is H\"{o}lder continuous in the ''weak'' (logarithmic) sense with the exponent power $\alpha=1/2.$ Moreover, it belongs to the class $W^{1, 2}_{\rm loc}.$ Under certain additional requirements, it can also be chosen as a homeomorphic solution. We give an appropriate example of the equation that satisfies all the conditions of the main result of the article, but does not have a homeomorphic Sobolev solution.

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.

On Dirichlet problem for Beltrami equations in finitely connected domains

Boundary value problems for the Beltrami equations are due to the famous Riemann dissertation (1851) in the simplest case of analytic functions and to the known works of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy--Riemann system. Of course, the Dirichlet problem was well studied for uniformly elliptic systems, see, e.g., \cite{Boj} and \cite{Vekua}. Moreover, the corresponding results on the Dirichlet problem for degenerate Beltrami equations in the unit disk can be found in the monograph \cite{GRSY}. In our article \cite{KPR1}, see also \cite{KPR3} and \cite{KPR5}, it was shown that each generalized homeomorphic solution of a Beltrami equation is the so-called lower $Q-$homeomorphism with its dilatation quotient as $Q$ and developed on this basis the theory of the boundary behavior of such solutions. In the next papers \cite{KPR2} and \cite{KPR4}, the latter made possible us to solve the Dirichlet problem with continuous boundary data for a wide circle of degenerate Beltrami equations in finitely connected Jordan domains, see also [\citen{KPR5}--\citen{KPR7}]. Similar problems were also investigated in the case of bounded finitely connected domains in terms of prime ends by Caratheodory in the papers [\citen{KPR9}--\citen{KPR10}] and [\citen{P1}--\citen{P2}]. Finally, in the present paper, we prove a series of effective criteria for the existence of pseudo\-re\-gu\-lar and multi-valued solutions of the Dirichlet problem for the degenerate Beltrami equations in arbitrary bounded finitely connected domains in terms of prime ends by Caratheodory.

On the Hilbert boundary-value problem for Beltrami equations with singularities

We investigate the Hilbert boundary-value problem for Beltrami equations $\overline\partial f=\mu\partial f$ with singularities in generalized quasidisks $D$ whose Jordan boundary $\partial D$ consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves. Generally speaking, generalized quasidisks do not satisfy the standard $(A)-$condition in PDE by Ladyzhenskaya-Ural'tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring-Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations. In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain $D$ of a general nature. As usual, a point in $\overline D$ is called a singularity of the Beltrami equation, if the dilatation quotient $K_{\mu}:=(1+|\mu|)/(1-|\mu|)$ is not essentially bounded in all its neighborhoods. Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincar\'{e} boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.