On the solvability of some nonclassical boundary-value problem for the Laplace equation in the plane corner (Q944163)

The author considers a non stationary boundary value problem for the Poisson equation with a dynamic boundary condition on the one part of the corner and a Dirichlet condition on the other part. The interest in this problem is because of the free boundary problem for the Laplace equation, which is the Hele--Shaw problem, in the case of free and fixed boundaries with corners at the initial time. The author follows a technique of reduction to a problem in a fixed domain. Then, differentiation leads to a non linear system of PDEs for which a one-value solvability must be achieved. Then the initial problem turns into a fixed point problem for a non linear operator. The main analytical problem is to show a weighted estimate in Hölder classes. This will be a starting point to study classical solvability of the free boundary problem related to the problem under consideration.