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On a Dirichlet Problem for One Improperly Elliptic Equation
Abstract
The Dirichlet problem for sixth order improperly elliptic equation is considered. The functional class of boundary functions, where this problem is normally solvable is determined. If the roots of the characteristic equation satisfy some conditions, the number of linearly independent solutions of homogeneous problem and the number of linearly independent solvability conditions of in-homogeneous problem are obtained. Solutions of homogeneous problem and solvability conditions of in-homogeneous problem are obtained in explicit form.
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Further mixed boundary value problems are studied for the inhomogeneous polyanalytic equation in the unit disc. As in part I, see [4], the boundary conditions are some combinations of Schwarz, Dirichlet and Neumann conditions. The method applied is based on an iteration process leading from lower order equations to higher ones, see [1-3]. The method can be used to solve all kind of such mixed problems also for other kinds of complex model equations.
We consider a strictly hyperbolic first-order system of three equations with constant coefficients in a bounded piecewise-smooth domain. The boundary of the domain is assumed to consist of six smooth noncharacteristic arcs. A boundary-value problem in this domain is posed by alternately prescribing one or two linear combinations of the components of the solution on these arcs. We show that this problem has a unique solution under certain additional conditions on the coefficients of these combinations, the boundary of the domain and the behaviour of the solution near the characteristics passing through the corner points of the domain. © 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.
* Basic Properties of Harmonic Functions * Bounded Harmonic Functions * Positive Harmonic Functions * The Kelvin Transform * Harmonic Polynomials * Harmonic Hardy Spaces * Harmonic Functions on Half-Spaces * Harmonic Bergman Spaces * The Decomposition Theorem * Annular Regions * The Dirichlet Problem and Boundary Behavior * Volume, Surface Area, and Integration on Spheres * Harmonic Function Theory and Mathematica * References * Symbol Index * Index
The three basic boundary value problems in complex analysis are of Schwarz, of Dirichlet and of Neumann type. When higher order equations are investigated all kind of combinations of these boundary conditions are proper to determine solutions. However, not all of these conditions are leading to well-posed problems. Some are overdetermined so that solvability conditions have to be found. Some of these boundary value problems are treated here for the inhomogeneous polyanalytic equation.
The paper considers the properly elliptic equation in multiply connected domains. An effective solution of Dirichlet problem is proposed by reduction to Fredholm integral equation of the second kind. Conditions ensuring unique solvability are derived.
The paper studies unique solvability in the open unit disk of the Dirichlet problem for a properly elliptic equation of order 2n in the class of 2n times continuously differentiate functions, which with up to n-th order derivatives satisfy the Hölder condition in the closed disk. The necessary and sufficient conditions for unique solvability are found.
A general (not necessarily local) boundary value problem is considered for an elliptic [] system on the plane of [ n]th order containing only leading terms with constant coefficients. By a method of function theory developed for elliptic [] systems of first order
[]
with a constant triangular matrix [], []; this problem is reduced to an equivalent system of integrofunctional equations on the boundary. In particular, a criterion that the problem be Noetherian and a formula for its index are obtained in this way. All considerations are carried out in the smooth case when the boundary of the domain has no corner points, while the boundary operators act in spaces of continuous functions.
The paper studies the unique solvability of the Dirichlet problem for some class of higher order properly elliptic equations. The different forms of necessary and sufficient conditions rendering the corresponding problem in being uniquely solvable and some applications are found.
A general boundary value problem, encompassing from a unified viewpoint a broad circle of local and nonlocal boundary value problems, is studied for elliptic systems with real, constant (and only leading) matrix coefficients. A method is given for the equivalent reduction of this problem to a system of boundary equations. The considerations are carried out in domains with piecewise smooth boundaries and in weighted spaces. A Noetherian criterion and an index formula for this problem are established, and the asymptotics of its solution in a neighborhood of corner points is described.
In this paper, we describe how to analyze boundary value problems for third-order nonlinear ordinary differential equations over an infinite interval. Several physical problems of interest are governed by such systems. The seminumerical schemes described here offer some advantages over solutions obtained by using traditional methods such as finite differences, shooting method, etc. These techniques also reveal the analytic structure of the solution function. For illustrative purposes, several physical problems, mainly drawn from fluid mechanics, are considered; they clearly demonstrate the efficiency of the techniques presented here.
We find the spectrum and prove a theorem on the expansion of an arbitrary function satisfying certain smoothness conditions in terms of the root functions of a boundary value problem of the type –y+q(x)+a/x2y=y, y(0)=0, M() y(a)+N() y(b)=0, where 0,0, M() and N() are polynomials with complex coefficients, and q(x) is a sufficiently smooth complex-valued function.
A new proof, based on the Perron-Frobenius theory of nonnegative matrices, is given of a result of Hurwitz on the sharpness of the classical Eneström-Kakeya theorem for estimating the moduli of the zeros of a polynomial with positive real coefficients. It is then shown (Theorem 2) that the zeros of a particular set of polynomials fill out the Eneström-Kakeya annulus in a precise manner, and this is illustrated by numerical results in Fig. 1.
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