A singularly perturbed nonlinear traction boundary value problem for linearized elastostatics. A functional analytic approach

Abstract

In this paper, we consider two bounded open subsets of Ωⁱ and Ω^o of Rⁿ containing 0 and a (nonlinear) function G^o of ∂Ω^o×Rⁿ to Rⁿ, and a map T of ]1-(2/n),+∞[ times the set M_n(R) of n× n matrices with real entries to M_n(R), and we consider the problem

div (T(ω,Du))=0  in Ω^oεclΩⁱ,

{-T(ω,Du)ν_εΩⁱ=0 on ε∂Ωⁱ,

(ω,Du(x))ν^o(x)=G^o(x,u(x)) ∀ x∈∂Ω^o,

where ν_εΩⁱ and ν^o denote the outward unit normal to ε∂Ωⁱ and ∂Ω^o, respectively, and where ε>0 is a small parameter. Here (ω-1) plays the role of ratio between the first and second Lamé constants, and T(ω,·) plays the role of (a constant multiple of) the linearized Piola Kirchhoff stress tensor, and G^o plays the role of (a constant multiple of) a traction applied on the points of ∂Ω^o. Then we prove that under suitable assumptions the above problem has a family of solutions {u(ε,·)}_{ε∈]0,ε´[} for ε´ sufficiently small and we show that in a certain sense {u(ε,·)}_{ε∈]0,ε´[} can be continued real analytically for negative values of ε.