On representation of Green's function of the Dirichlet problem for biharmonic equation in a ball

V. V. Karachik

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 4, pp. 13-22

Résumé

Elementary solution of a biharmonic equation is introduced in analogy to the known elementary solution of the Laplace equation. Relation of this elementary solution with the elementary solution of the Laplace equation gets determined. Depending on dimensionality of space in which a boundary problem is being under research, a symmetric function of two variables gets determined in explicit form through the introduced elementary solution. Then it gets proved that this function possesses properties of Green's function of the Dirichlet problem for biharmonic equation in a unit ball. Two cases when space dimensionality equals two and when space dimensionality is more than two are being researched separately. Analogous to Green's function of the Dirichlet problem for Poisson's equation in a ball, there is expansion of Green's function of the Dirichlet problem for biharmonic equation in a ball in the full, orthogonal-at-the-unit-sphere, system of homogenous harmonic polynominals. This is to be done in case when space dimensionality is more than four. Using the obtained expansion of Green's function, integral gets calculated by a ball with the kernel out of Green's function from a homogenous harmonic polynominal multiplied by the positive degree of norm of the independent variable. The obtained results get complied with the previously known results in this sphere.

Détail

Keywords: Dirichlet problem, biharmonic equation, Green's function.

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