Recent zbMATH articles in MSC 30E25https://www.zbmath.org/atom/cc/30E252022-05-16T20:40:13.078697ZWerkzeugAnalog of the Hadamard theorem and related extremal problems on the class of analytic functionshttps://www.zbmath.org/1483.300012022-05-16T20:40:13.078697Z"Akopyan, R. R."https://www.zbmath.org/authors/?q=ai:akopyan.roman-razmikovichSummary: We study several related extremal problems for analytic functions in a finitely connected domain \(G\) with rectifiable Jordan boundary \(\Gamma \). A sharp inequality is established between values of a function analytic in \(G\) and weighted means of its boundary values on two measurable subsets \(\gamma_1\) and \(\gamma_0=\Gamma\setminus\gamma_1\) of the boundary:
\[ |f(z_0)|\leq\mathcal{C}\,\|f\|^{\alpha}_{L^q_{\varphi_1}(\gamma_1)}\, \|f\|^{\beta}_{L^p_{\varphi_0}(\gamma_0)},\quad z_0\in G,\quad 0<q,p\leq\infty.\]
The inequality is an analog of Hadamard's three-circle theorem and the Nevanlinna brothers' two-constant theorem. In the case of a doubly connected domain \(G\) and \(1\leq q,p\leq\infty \), we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from the part \(\gamma_1\) of the boundary to a given point of the domain. In these cases, the corresponding problem of optimal recovery of a function from its approximate boundary values on \(\gamma_1\) and the problem of the best approximation of a functional by bounded linear functionals are solved. The case of a simply connected domain \(G\) has been completely investigated previously.Regularization of a class of summary equationshttps://www.zbmath.org/1483.300702022-05-16T20:40:13.078697Z"Garif'yanov, F. N."https://www.zbmath.org/authors/?q=ai:garifyanov.farkhat-nurgayazovich"Strezhneva, E. V."https://www.zbmath.org/authors/?q=ai:strezhneva.elena-vasilevnaSummary: Let \(D\) be an arbitrary quadrangle with boundary \(\Gamma \). We consider a four-element linear summary equation. The solution is sought in the class of functions which are holomorphic outside \(D\) and vanish at infinity. The boundary values satisfy the Hölder condition on any compact set which does not contain the vertices. At the vertices, singularities at most of logarithmic order are allowed. The coefficients of the equation are holomorphic in \(D\) and their boundary values satisfy the Hölder condition on \(\Gamma \). The free term satisfies the same conditions. The solution is sought in the form of the Cauchy type integral over \(\Gamma\) with unknown density. To regularize the obtained functional equation, we use the Carleman problem. Previously, a Carleman shift is introduced on \(\Gamma \); it transfers each side to itself and reverses orientation; the midpoints of the sides are fixed under the shift. We indicate some applications of this summary equation to the problem of moments for entire functions of exponential type.The Dirichlet-Neumann boundary value problem for the inhomogeneous Bitsadze equation in a ring domainhttps://www.zbmath.org/1483.300712022-05-16T20:40:13.078697Z"Gençtürk, İlker"https://www.zbmath.org/authors/?q=ai:gencturk.ilkerSummary: In this study, by using some integral representations formulas, we study solvability conditions and explicit solution of the Dirichlet-Neumann problem, an example for a combined boundary value problem, for the Bitsadze equation in a ring domain.Polyanalytic boundary value problems for planar domains with harmonic Green functionhttps://www.zbmath.org/1483.300722022-05-16T20:40:13.078697Z"Begehr, Heinrich"https://www.zbmath.org/authors/?q=ai:begehr.heinrich"Shupeyeva, Bibinur"https://www.zbmath.org/authors/?q=ai:shupeyeva.bibinurThe authors characterize the solvability of three boundary value problems for the inhomogeneous polyanalytic equation in planar domains (having a harmonic Green function), namely the well-posed iterated Schwarz problem, and two over-determined iterated problems of Dirichlet and Neumann type. Solutions formulas are also obtained and, in particular, it is concluded that the polyanalytic Cauchy-Pompeiu representation formula provides the solution to the Dirichlet problem (for any degree \(n\), and in the cases for which the solution exists).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)Asymptotics of the Riemann-Hilbert problem for the Somov model of magnetic reconnection of long shock waveshttps://www.zbmath.org/1483.300732022-05-16T20:40:13.078697Z"Bezrodnykh, S. I."https://www.zbmath.org/authors/?q=ai:bezrodnykh.sergei-i"Vlasov, V. I."https://www.zbmath.org/authors/?q=ai:vlasov.vladimir-ivanovichSummary: We consider the Riemann-Hilbert problem in a domain of complicated shape (the exterior of a system of cuts), with the condition of growth of the solution at infinity. Such a problem arises in the Somov model of the effect of magnetic reconnection in the physics of plasma, and its solution has the physical meaning of a magnetic field. The asymptotics of the solution is obtained for the case of infinite extension of four cuts from the given system, which have the meaning of shock waves, so that the original domain splits into four disconnected components in the limit. It is shown that if the coefficient in the condition of growth of the magnetic field at infinity consistently decreases in this case, then this field basically coincides in the limit with the field arising in the Petschek model of the effect of magnetic reconnection.Noether property and approximate solution of the Riemann boundary value problem on closed curveshttps://www.zbmath.org/1483.300742022-05-16T20:40:13.078697Z"Bory-Reyes, Juan"https://www.zbmath.org/authors/?q=ai:bory-reyes.juan|moreno-garcia.tania"Katz, David"https://www.zbmath.org/authors/?q=ai:katz.david-f|katz.david-bBased on classic methods, the authors analyse the Noether property of a Riemann boundary value problem in the Banach algebra of continuous functions over closed curves, as well as consequent approximate solutions (by using quasi-Fredholm operators).
Reviewer: Luis Filipe Pinheiro de Castro (Aveiro)On the splitting type of holomorphic vector bundles induced from regular systems of differential equationhttps://www.zbmath.org/1483.300752022-05-16T20:40:13.078697Z"Giorgadze, Grigori"https://www.zbmath.org/authors/?q=ai:giorgadze.grigory"Gulagashvili, Gega"https://www.zbmath.org/authors/?q=ai:gulagashvili.gegaSummary: We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.On the number of linearly independent solutions of the Riemann boundary value problem on the Riemann surface of an algebraic functionhttps://www.zbmath.org/1483.300762022-05-16T20:40:13.078697Z"Kruglov, V. E."https://www.zbmath.org/authors/?q=ai:kruglov.vladislav-e|kruglov.viktor-eSummary: We suggest a modified solution to the Riemann boundary value problem on a Riemann surface of an algebraic function of genus \(\rho \). This allows us to to reduce the problem of finding the number \(l\) of linearly independent algebraic functions (LIAF), that are multiples of a fractional divisor \(Q\), to finding the number of LIAF that are multiples of an effective divisor \(J \) (\(\operatorname{ord}J = \rho\)); this provides a solution to the Jacobi inversion problem given in this paper. We study the case, where the exponents of the normal basis elements coincide, and solve the problem of finding the number of LIAF, multiples of an effective divisor. The definitions of conjugate points of Riemann surface and hyperorder of an effective divisor are introduced. Depending on the structure of divisor \(J\), exact formulas are obtained for number \(l\); they are expressed in terms of the order of divisor \(Q\), the hyperorder of divisor \(J\), and numbers \(\rho\) and \(n\), where \(n\) is the number of sheets of the algebraic Riemann surface.Free boundary problems in the spirit of Sakai's theoremhttps://www.zbmath.org/1483.300772022-05-16T20:40:13.078697Z"Vardakis, Dimitris"https://www.zbmath.org/authors/?q=ai:vardakis.dimitris"Volberg, Alexander"https://www.zbmath.org/authors/?q=ai:volberg.alexander-lSummary: A Schwarz function on an open domain \(\Omega\) is a holomorphic function satisfying \(S(\zeta)=\bar{\zeta}\) on \(\Gamma\), which is part of the boundary of \(\Omega\). \textit{M. Sakai} [Acta Math. 166, No. 3-4, 263--297 (1991; Zbl 0728.30007)] gave a complete characterization of the boundary of a domain admitting a Schwarz function. In fact, if \(\Omega\) is simply connected and \(\Gamma =\partial\Omega\cap D(\zeta_0,r)\), then \(\Gamma\) has to be regular real analytic (with possible cusps). Sakai's result has natural applications to 1) quadrature domains, 2) free boundary problem for \(\Delta u=1\) equation. In our scenarios \(\Gamma\) can be, respectively, from real-analytic to just \(C^\infty\), regular except for a harmonic-measure-zero set, or regular except finitely many points.On the investigation of isotropic thick-walled shellshttps://www.zbmath.org/1483.300902022-05-16T20:40:13.078697Z"Khvoles, A."https://www.zbmath.org/authors/?q=ai:khvoles.a-r|khvoles.alexander|khvoles.a-a"Zgenti, V."https://www.zbmath.org/authors/?q=ai:zgenti.v"Vashakmadze, T."https://www.zbmath.org/authors/?q=ai:vashakmadze.tamaz-s|vashakmadze.tamara-sSummary: We consider the problems of creating 2-dim models for thin-walled structures and satisfaction of boundary conditions when the generalized stress vector is given on the surfaces for elastic plates and shells. This problem was open also both for refined theories in the wide sense and hierarchical type models.A note on the phase retrieval of holomorphic functionshttps://www.zbmath.org/1483.300972022-05-16T20:40:13.078697Z"Perez, Rolando III"https://www.zbmath.org/authors/?q=ai:perez.rolando-iiiSummary: We prove that if \(f\) and \(g\) are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then \(f=g\) up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of \(\pi\). We also prove that if \(f\) and \(g\) are functions in the Nevanlinna class, and if \(|f|=|g|\) on the unit circle and on a circle inside the unit disc, then \(f=g\) up to the multiplication of a unimodular constant.