A boundary value problem for the bi-harmonic equation and the iterated Laplacian in a 3d-domain with an edge∗

For domains Ω with piecewise smooth boundaries the generalized solution u ∈ W 2 2 (Ω) of the equation ∆xu = f with the boundary conditions u = ∆xu = 0 in general cannot be obtained by solving iteratively a system of two Poisson equations under homogeneous Dirichlet conditions. Such a system is obtained by setting v = −∆u. In the two-dimensional case this fact is known as the Sapongyan paradox in the theory of simply supported polygonal plates. In the present paper the three-dimensional problem is investigated for a domain with a smooth edge Γ. If the variable opening angle α ∈ C∞(Γ) is less than π everywhere on the edge, the boundary value problem for the bi-harmonic equation is equivalent to the iterated Dirichlet problem and its solution u inherits the positivity preserving property from these problems. In the case that α ∈ (π, 2π) the procedure of solving the two Dirichlet problems must be modified by permitting an infinite-dimensional kernel and co-kernel of operators and determining the solution u ∈ W 2 2 (Ω) through inverting a certain integral operator on the contour Γ. If α(s) ∈ (3π/2, 2π) for a point s ∈ Γ then there exists a non-negative function f ∈ L2(Ω) for which the solution u changes sign inside the domain Ω. In the case of the crack, that is (α = 2π everywhere on Γ), one needs to introduce a special scale of weighted function spaces. Also there the positivity preserving property fails. In some geometrical situations the questions of a correct setting for the boundary value problem of the bi-harmonic equation and the positivity property remain open.