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

Renardy, Michael

Differ. Integral Equ. 15, No. 10, 1273-1279 (2002).

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.

Reviewer: Valery V.Karachik (Tashkent)

MSC:

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

Keywords:

Neumann problem; Laplace’s equation; two dimensions; Fredholm index