Microscopically weakly singularly perturbed loads for a nonlinear traction boundary value problem: a functional analytic approach

Let Omega(i) and Omega(o) be two bounded open subsets of R(n) containing 0. Let G(i) be a (nonlinear) map of partial derivative Omega(i) x R(n) to R(n). Let a(o) be a map of partial derivative Omega(o) to the set M(n)(R) of n x n matrices with real entries. Let g be a function of partial derivative Omega(o) to R(n). Let gamma be a positive valued function defined on a right neighbourhood of 0 on the real line. Let T be a map of] 1 - (2/n), + infinity[xM(n)(R) to M(n)(R). Then we consider the problem {divdT(omega, Du)) = 0 in Omega(o) backslash epsilon c1 Omega(i), -T(omega, Du(x))v(epsilon Omega)(i)(x) = 1/gamma(epsilon) G(i)(x.epsilon, u(x)) for all x is an element of epsilon partial derivative Omega(i), T(omega, Du(x))v(o)(x) = a(o)(x)u(x) + g(x) for all x is an element of partial derivative Omega(o) where nu(epsilon)Omega(i) and nu(o) denote the outward unit normal to epsilon partial derivative Omega(i) and partial derivative Omega(o), respectively, and where epsilon > 0 is a small parameter. Here (omega - 1) plays the role of ratio between the first and second Lame constants and T(omega, .) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor. Under the condition that lim(epsilon -> 0) gamma(-1) (epsilon)epsilon(log epsilon)(delta 2,n) exists in R, we prove that under suitable assumptions the above problem has a family of solutions {u(epsilon, .)}(epsilon epsilon]0,epsilon'[) for epsilon' sufficiently small and we analyse the behaviour of such a family as epsilon approaches 0 by an approach which is alternative to those of asymptotic analysis. Here delta(2, n) denotes the Kronecker symbol.