The Neumann problem with boundary condition on an open plane surface (Q1874779)

Given a convex open domain \(\Sigma \subset \mathbb{R}^2 \subset \mathbb{R}^3\) whose boundary is a piecewise-smooth closed curve \(L= \partial \Sigma \); the author consids the problem of finding a harmonic function in \(\mathbb{R}^3 \setminus ( \Sigma VL)\) with the Neumann condition on \(\Sigma\): \[ \left( \frac{\partial u}{\partial n} \right) ^+ = \left( \frac{\partial u}{\partial n} \right) ^- = f . \] Both classical and generalized (i.e., distributional) solutions of the above problem are treated. Section 2 provides the (sufficient) conditions ensuring the uniqueness of a strong generalized solution and describes the function classes to which the solution belongs for certain classes of the right-hand sides \(f\). Section 3 considers the case of \(\Sigma\) being the open unit disc. The solution is looked for in the form of a double layer potential which leads to the study of several integral operators; it is concluded that given a right-hand side \(f\) in a Hölder-type weighted space, then there exists another, in general, Hölder-type weighted space which contains a (unique) solution. In Section 4 there are constructed some auxiliary solutions for the case of the open unit disc which are used in Section 5 for constructing classical solutions for the Neumann problem in the general case of a piecewise-smooth closed boundary \(L\).