Boundary Problems for the Biharmonic Operator in a Square with $L^P $-Data

Abstract

Applied boundary value problems for the bi-Laplacian often actually involve domains with nonsmooth boundaries. The second fundamental problem of plane elasticity, the viscous flow and Stokes problem are representative examples of problems which can be reduced to the interior or exterior Dirichlet problem for the bi-Laplacian. In this paper we study the Dirichlet problem for $\Delta ^2 $ in a plane square with $L^p $-data, looking for a solution in potential form. By the method of pseudodifferential operators we show an existence theorem provided the data are in $L^p $, $1 < p < 3$, and satisfy a proper compatibility condition. We also solve two further boundary value problems which are related to the Dirichlet problem.