A nonlinear problem for the Laplace equation with a degenerating Robin condition

We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain \u3a9. On a part of the boundary 02\u3a9, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter \u3b4 in such a way that for \u3b4 = 0 it degenerates into a Neumann condition. For \u3b4 small and positive, we prove that the boundary value problem has a solution u(\u3b4,\ub7). We describe what happens to u(\u3b4,\ub7) as \u3b4\u21920 by means of representation formulas in terms of real analytic maps. Then, we confine ourselves to the linear case, and we compute explicitly the power series expansion of the solution