Asymptotic behaviour of the solutions of a non-linear transmission problem for the Laplace operator in a domain with a small hole. A functional analytic approach

We consider a bounded open subset I-o of R-n with 0 is an element of I-o, and a function f(o) of partial derivative I-o to R. Under reasonable assumptions, the Dirichlet problem Delta u = 0 in I-o, u = f(o) on partial derivative I-o, has one and only one solution (u) over tilde (o). Then we consider another bounded open subset I-i of R-n with 0 is an element of I-i, and an increasing diffeomorphism F of R onto itself, and a constant gamma is an element of]0, +infinity[, and a function g of partial derivative I-i to R, and we consider the non- linear transmission boundary value problem Delta u(i) = 0 in is an element of I-i Delta u(o) = 0 in I-o \ epsilon clI(i), u(o) = F(u(i)) on epsilon partial derivative I-i, partial derivative u(o/)partial derivative v(is an element of Ii) (x ) = gamma partial derivative mu i/partial derivative v epsilon(i)(I)(x) + g(x/epsilon) for all x is an element of epsilon dI(i), u(0) = f(0) on partial derivative I-o, for epsilon > 0 small, where nu(epsilon Ii) is the outward unit normal to epsilon partial derivative(i)(I). Under suitable conditions on the data, we show that for (epsilon) over tilde > 0 sufficiently small, such a boundary value problem admits locally around (F((-1))((mu) over bar (o)(0)), (mu) over bar (sigma)) a family of solutions {(ui(epsilon,.), uo(epsilon,.))}epsilon is an element of]0,(epsilon) over bar[. Then we show that u(i)(epsilon, epsilon.) and (suitable restrictions of) u(o) (epsilon, .) and u(o) (epsilon, epsilon.) can be continued real analytically in the parameter epsilon around epsilon = 0 for n >= 3, and can be represented in terms of real analytic functions of epsilon, log(-1) epsilon, epsilon log(2) epsilon for n = 2.