Search

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., Comment: 31 pages

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., Comment: 26 pages. arXiv admin note: text overlap with arXiv:1511.02798

Applying the known Nyman--Beurling criterion, it is disproved the Riemann hypothesis on zeros of $\zeta -$function., Comment: 9 pages

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 $u=U\circ\omega,$ where $\omega(z)$ 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., Comment: 30 pages

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 $2\pi -$periodic bounded integrands that are differentiable a.e. and, in particular, for integrands of the class ${\cal {CBV}}$ (countably bounded variation)., Comment: 18 pages

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., Comment: 35 pages

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., Comment: 8 pages

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., Comment: 29 pages

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, Comment: 24 pages, in English

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., Comment: 31 pages. arXiv admin note: substantial text overlap with arXiv:1510.06380

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., Comment: 8 pages

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., Comment: 5 pages

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., Comment: 7 pages

We show that every homeomorphic $W^{1,1}_{\rm loc}$ solution $f$ of a Beltrami equation $\overline{\partial}f=\mu\,\partial f$ in a domain $D\subseteq\Bbb C$ is the so--called ring $Q-$homeomorphism with $Q(z)=K^T_{\mu}(z, z_0)$ where $K^T_{\mu}(z, z_0)$ 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 $\overline{\partial}f=\mu\,\partial f$, 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 $\varphi$ that are continuous with respect to the topology of prime ends., Comment: 32 pages, 50 references. arXiv admin note: substantial text overlap with arXiv:1502.01603, arXiv:1210.5910, arXiv:1201.5570, arXiv:1503.04306

It is developed the theory of the boundary behavior of homeomorphic solutions of the Beltrami equations ${\bar{\partial}}f=\mu\,{\partial}f$ 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 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., Comment: 19 pages, 20 references

In the present paper, it was 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 developed theory is applied to mappings with finite distortion by Iwaniec, also to solutions of the Beltrami equations, as well as to finitely bi--Lipschitz mappings that a far--reaching extension of the known classes of isometric and quasiisometric mappings., Comment: 30 pages. arXiv admin note: substantial text overlap with arXiv:1408.0470, arXiv:1401.4839, arXiv:1210.5910

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 $K_I(x,f)$ is its inner dilatation. The statement is valid also for all finitely bi-Lipschitz mappings that a far--reaching extension of the well-known classes of isometric and quasiisometric mappings. This makes pos\-sib\-le to apply our theory of the boundary behavior of the lower $Q$-homeomorphisms to all given classes., Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1401.4839, arXiv:1012.5010

It is proved that the dimension of the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. is infinite., Comment: 5 pages

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., Comment: 29 pages

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., Comment: 40 pages, we improve modulus estimates and on this basis prove a series of new criteria for homeomorphic extension of spatial mappings to the boundary by prime ends in terms of inner dilatations

With no criteria of the index type, it is proved the existence of a solution for the Riemann-Hilbert problem in the fairly general setting of arbitrary Jordan domains, 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 arbitrary rectifiable boundaries stated in terms of the natural parameter and nontangential limits. Moreover, it is shown that the dimension of the spaces of solutions is infinite., Comment: 10 pages

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 theory of the boundary behavior of such solutions. Then, on this basis, we show that, for wide classes of degenerate Beltrami equations $\bar{\partial}f=\mu \partial f$, there exist regular solutions of the Dirichlet problem in arbitrary Jordan domains in $\Bbb C$ and pseudoregular and multi-valued solutions in arbitrary finitely connected domains in $\Bbb C$ bounded by mutually disjoint Jordan curves., Comment: 35 pages. arXiv admin note: substantial text overlap with arXiv:1201.5570

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 Orlicz--Sobolev homeomorphisms and on semicontinuity in the mean of dilatations of the Sobolev homeomorphisms. These results lead us to closeness of the corresponding classes of homeomorpisms. Finally, we come on this basis to criteria of compactness of classes of Sobolev's homeomorphisms with the corresponding conditions on dilatations and two fixed points., Comment: 47 pages, precision

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 that the function $Q$ is of finite mean oscillation. These results will have wide applications to Sobolev's mappings., Comment: 23 pages

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 the Beltrami equations in arbitrary Jordan domains and in arbitrary finitely connected domains bounded by mutually disjoint Jordan curves, correspondingly., Comment: 28 pages, 2 figures

We construct variations for the classes of regular solutions to degenerate Beltrami equations with restrictions of the set-theoretic type for the complex coefficient. On this basis, we prove the variational maximum principle and other necessary conditions of extremum., Comment: 13 pages

First of all, we prove that open mappings in Orlicz-Sobolev classes $W^{1,\phi}_{\rm loc}$ under the Calderon type condition on $\phi$ have the total differential a.e. that is a generalization of the well-known theorems of Gehring-Lehto-Menchoff in the plane and of V\"ais\"al\"a in ${\Bbb R}^n$, $n\geqslant3$. Under the same condition on $\phi$, we show that continuous mappings $f$ in $W^{1,\phi}_{\rm loc}$, in particular, $f\in W^{1,p}_{\rm loc}$ for $p>n-1$ have the $(N)$-property by Lusin on a.e. hyperplane. Our examples demonstrate that the Calderon type condition is not only sufficient but also necessary for this and, in particular, there exist homeomorphisms in $W^{1,n-1}_{\rm loc}$ which have not the $(N)$-property with respect to the $(n-1)$-dimensional Hausdorff measure on a.e. hyperplane. It is proved on this base that under this condition on $\phi$ the homeomorphisms $f$ with finite distortion in $W^{1,\phi}_{\rm loc}$ and, in particular, $f\in W^{1,p}_{\rm loc}$ for $p>n-1$ are the so-called lower $Q$-homeomorphisms where $Q(x)$ is equal to its outer dilatation $K_f(x)$ as well as the so-called ring $Q_*$-homeomorphisms with $Q_*(x)=[K_{f}(x)]^{n-1}$. This makes possible to apply our theory of the local and boundary behavior of the lower and ring $Q$-homeomorphisms to homeomorphisms with finite distortion in the Orlicz-Sobolev classes., Comment: 69 pages, 2 figures, In Memory of Alberto Calderon (1920-1998)