The simple layer potential for the biharmonic equation in n variables (Q810679)

Let \(\Omega\) be a bounded Lyapunov domain in \({\mathbb{R}}^ n\) and let \(g_ 0,g_ 1,...,g_ n\in C^{1+\lambda}(\partial \Omega)\). The author gives necessary and sufficient conditions for the existence of a function \(u\in C^{2+\lambda}({\bar \Omega})\) such that u is biharmonic on \(\Omega\) and satisfies the boundary conditions \(u|_{\partial \Omega}=g_ 0\) and \((\partial u/\partial x_ j)|_{\partial \Omega}=g_ j\) \((j=1,...,n)\). He also shows that when u exists it is unique, and gives a representation for u which involves an integral over \(\partial \Omega\), called a simple layer potential. The work depends upon a study of a new class of singular integral operators, each taking an n- tuple of scalar-valued functions into an n-tuple of differential forms of degree 1.