Mixed boundary condition

In mathematics, a mixed boundary condition for a partial differential equation defines a boundary value problem in which the solution of the given equation is required to satisfy different boundary conditions on disjoint parts of the boundary of the domain where the condition is stated.

For example, given a solution u to a partial differential equation on a domain Ω with boundary ∂Ω, it is said to satisfy a mixed boundary condition if, consisting ∂Ω of two disjoint parts, Γ1 and Γ2, such that ∂Ω = Γ1 ∪ Γ2, u verifies the following equations: where u0 and g are given functions defined on those portions of the boundary.

M. Wirtinger, dans une conversation privée, a attiré mon attention sur le probleme suivant: déterminer une fonction u vérifiant l'équation de Laplace dans un certain domaine (D) étant donné, sur une partie (S) de la frontière, les valeurs périphériques de la fonction demandée et, sur le reste (S′) de la frontière du domaine considéré, celles de la dérivée suivant la normale.

Je me propose de faire connaitre une solution très générale de cet intéressant problème.

[2]The first boundary value problem satisfying a mixed boundary condition was solved by Stanisław Zaremba for the Laplace equation: according to himself, it was Wilhelm Wirtinger who suggested him to study this problem.