Integral Formulations for two-dimensional Multi-Arcs

Abstract

We study the Laplace equation with Dirichlet and Neumann boundary conditions posed on multi-arcs, i.e., collections of open arcs meeting at junction points. We begin by introducing a scale of Sobolev spaces constructed using the Sobolev spaces on open arcs as main building block and extend the definition of trace operators. We reformulate the boundary value problems using boundary integral formulations. We then establish a well-posed integral formulation for the Dirichlet problem, which can be discretized using standard numerical methods. We further investigate the singular behavior of the solution densities at branch points through numerical experiments and observe that these singularities are comparable to the corner singularities arising in polygonal domains. For the Neumann problem, we show that the associated hypersingular operator is not necessarily invertible on classical Sobolev spaces and provide numerical evidence that solutions may develop jump discontinuities at branch points.