Boundary value problems for the bi-harmonic equation and for the iterated Laplacian in a three-dimensional domain with an edge

S. A. Nazarov ; G. H. Sweers

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 37, Tome 336 (2006), pp. 153-198

Résumé

For domains $\Omega$ with piecewise smooth boundaries the generalized solution $u\in W^2_2(\Omega)$ of the equation $\Delta_x^2u=f$ with the boundary conditions $u=\Delta_xu=0$ 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 $v=-\Delta u$. 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 $\Gamma$. If the variable opening angle $\alpha\in C^\infty(\Gamma)$ is less than $\pi$ 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 $\alpha\in(\pi,2\pi)$ 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 $u\in W^2_2(\Omega)$ through inverting a certain integral operator on the contour $\Gamma$. If $\alpha(s)\in(3\pi/2,2\pi)$ for a point $s\in\Gamma$ then there exists a non-negative function $f\in L_2(\Omega)$ for which the solution $u$ changes sign inside the domain $\Omega$. In the case of the crack, that is ($\alpha=2\pi$ everywhere on $\Gamma$), 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.