On the solvability of some nonclassical boundary-value problem for the Laplace equation in the plane corner. (English) Zbl 1153.35092

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.


35R35 Free boundary problems for PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs