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A Dirichlet Problem for Non-elliptic Equations and Chebyshev Polynomials
Abstract
We consider the Dirichlet problem for the linear non-elliptic fourth order partial differential equation in the unit disk. It supposed that in the equation only fourth order terms and coefficients are constant. The solvability conditions of in-homogeneous problem and the solutions of the corresponding homogeneous problem are determined in explicit form. The solutions are obtained in the form of expansions by Chebyshev polynomials.
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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 the homogeneous problem and the number of linearly independent solvability conditions of the inhomogeneous problem are determined. Solutions of the homogeneous problem and solvability conditions of the inhomogeneous problem are obtained in explicit form.
The Dirichlet problem for a class of properly elliptic sixth-order equations in the unit disk is considered. Formulas for determining the defect numbers of this problem are obtained. Linearly independent solutions of the homogeneous problem and conditions for the solvability of the inhomogeneous problem are given explicitly.
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
For higher-order partial differential equations in two or three variables, the Dirichlet problemin rectangular domains is studied. Small denominators hampering the convergence of series appear in the process of constructing the solution of the problem by the spectral decomposition method. A uniqueness criterion for the solution is established. In the two-dimensional case, estimates justifying the existence of a solution of the Dirichlet problem are obtained. In the three-dimensional case where the domain is a cube, it is shown that the uniqueness of the solution of the Dirichlet problem is equivalent to the great Fermat problem.
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.
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