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 Ω. On a part of the boundary ∂Ω, we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter δ in such a way that for δ = 0 it degenerates into a Neumann condition. For δ small and positive, we prove that the boundary value problem has a solution u(δ,·). We describe what happens to u(δ,·) as δ→0 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.

We investigate the behavior of the solutions of a mixed problem for the Laplace equation in a domain . On a part of the boundary , we consider a Neumann condition, whereas in another part, we consider a nonlinear Robin condition, which depends on a positive parameter in such a way that for =0 it degenerates into a Neumann condition. For small and positive, we prove that the boundary value problem has a solution u(,). We describe what happens to u(,) as 0 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.