Higher order Neumann problems for Laplace’s equation in two dimensions. (English) Zbl 1017.35031

The boundary value problem \[ \left. \Delta u(x)=f(x),\quad x\in\Omega; \qquad \frac{\partial^ku}{\partial n^k}\right |_{\partial\Omega}=g(s),\quad s\in\partial\Omega\tag{*} \] in a bounded two-dimensional domain \(\Omega\subset\mathbb R^2\) is considered. The following main results are obtained. Let \(\Omega\) be a simply-connected domain with a smooth boundary \(\partial\Omega\), then every solution of the boundary value problem (*) is a harmonic polynomial of degree \(k-1\) (Theorem 1). For “generic” multiply-connected domains \(\Omega\) (in the \(C^m\)-topology with \(m>k\)) this is also true (Theorem 2). However, in domains with corners there are additional solutions.


35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35J25 Boundary value problems for second-order elliptic equations
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions