Closedness of the Biharmonic System of Basic Potentials

Abstract

The shift systems of the fundamental solution of biharmonic equation are considered. The shift sequence belongs to the complement of the bounded single-bond domain $Q$, so that all system functions satisfy the biharmonic equation in $Q$. The space of biharmonic functions $G_{2}(Q)$ is introduced as a closure in the $L_{2}(Q)$ norm of the linear shell of all kinds of shifts of the fundamental solution of biharmonic equation. The following problem is considered: what conditions the countable sequence of shifts must have so that the linear shell closure matches the entire $G _ {2} (Q) $ space, or, in other words, when is the considered system complete in $G _ {2} (Q)$? The problem for the harmonic case was considered by V.G. Lezhnev, he introduced a sufficient basis condition and proved the completeness and linear independence of shift systems of the fundamental solution of Laplace's equation.The work generalizes the basis condition to the biharmonic case and proves the closedness and linear independence of biharmonic system. The basis condition is related to the singularity condition of biharmonic functions. The proof relies on the property of continuity of potentials with a biharmonic nucleus and on the Rikier's boundary value problem solution.