Smooth potentials with prescribed boundary behaviour
Let \(D \subset \mathbb R^n\) be a \(C^1\) domain where \(n \geq 2\). The main problem here is to study whether, given a continuous function \(g:\partial D \to (0, +\infty)\), there exists a function \(v \in C^1(\overline{ D})\) which is superharmonic on \(D\) and satisfies the boundary conditions \( v(z)= 0\) and \(\partial v/ \partial n_z = g(z) \) for \(z \in \partial D\), where \(\partial / \partial n_z \) denotes differentiation in the direction of the inward normal at \(z\). The authors prove that this is the case for so-called Lyapunov-Dini domains. Membership in the Lyapunov-Dini class requires that the angles between the directions \(n_z\) vary slowly enough (Dini condition). An example is given to show that the result is nearly sharp.
scientific article; zbMATH DE number 2074338