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Defect Numbers of the Dirichlet Problem for a Properly Elliptic Sixth-Order Equation
Abstract
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 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.
The goal of this note is to consider the application of the complex interpolation space of Morrey spaces. Actually, the boundedness of Riesz potentials acting on Morrey spaces, which is obtained by Adams, is refined by means of the complex interpolation.
* 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
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