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

Abstract

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.

Interaction of scales for a singularly perturbed degenerating nonlinear Robin problem

“…Dalla Riva and Mishuris [17] have investigated the solvability of a small nonlinear perturbation of a homogeneous linear transmission problem by potential theoretical techniques. The present paper represents a continuation of the analysis done in [49], where the authors of the present paper have considered the behavior as δ → 0 of the solutions to the boundary value problem…”

“…As is well known, the Neumann problem above may have infinite solutions or no solutions, depending on compatibility conditions on the Neumann datum. In [49], we have proved that, under suitable assumptions, solutions to (1) exist and we have shown that they diverge if the compatibility condition on the Neumann datum for the existence of solutions to (2) does not hold. In [49], we have considered a Robin problem as simplified model for the transmission problem for a composite domain with imperfect (nonnatural) conditions along the joint boundary.…”

“…In [49], we have proved that, under suitable assumptions, solutions to (1) exist and we have shown that they diverge if the compatibility condition on the Neumann datum for the existence of solutions to (2) does not hold. In [49], we have considered a Robin problem as simplified model for the transmission problem for a composite domain with imperfect (nonnatural) conditions along the joint boundary. Such nonlinear transmission conditions frequently appear in practical applications for various nonlinear multiphysics problems (e.g., [9,42,43,44,45,46,47,53]).…”

“…In [49], we have considered the case where the surface where we consider the Robin condition is the boundary of a fixed hole Ω i . Here we wish to study the case where the hole becomes small and degenerates into a point.…”

“…However, in literature one can find examples where the geometry changes in a more drastic way, as, for example, the case of oscillating boundaries (see, e.g., [4,3,22]). On the other hand, one may also consider the case where the geometry is fixed and the boundary condition is changing as in [49].…”