Bernoulli’s Free-boundary Problem

As already mentioned in the introduction Bernoulli’s free-boundary problem arises in ideal fluid dynamics, optimal design, electro chemistry, electro statics, and further applications. In the interior Bernoulli problem a connected domain Ω in ℝ n and a constant Q > 0 are given. The task is to find a subset A⊂Ω and a potential u: Ω\A → ℝ such that $$ \begin{gathered} - \Delta u = 0 in \Omega \backslash A, \hfill \\ u = 0 on \partial \Omega , \hfill \\ u = 1 on \partial A, \hfill \\ \frac{{\partial u}} {{\partial v}} = Q on \partial A \hfill \\ \end{gathered} $$ (see Figure 1.1 on page 2). In the exterior Bernoulli problem ∂A is exterior to ∂Ω with u = 1 on ∂Ω, u = 0 on ∂A and Q < 0 (Figure 14.1). The same problem can be posed for the p-Laplacian (Acker and Meyer [3]). We mainly consider the semilinear problem. Typically the interior Bernoulli problem has two solutions, an elliptic one close to the fixed boundary, and a hyperbolic (low energy) solution far from the boundary. Hyperbolic solutions are more delicate for analysis and numerical approximation. Nevertheless there is a second order trial free-boundary method, the implicit Neumann scheme (Section 14.3), with equally good performance for both types of solutions. See our paper with M. Rumpf [56] for the convergence proof. To begin with we describe the applications in physics and industry in detail.