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 Ω i and Ω o of R n containing 0 and a (nonlinear) function G o of ∂Ω o ×R n to R n , 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 o ε clΩ i , {- T ( ω , Du ) ν ε Ω i =0 on ε ∂Ω i , (ω , Du ( x )) ν o ( x )= G o ( x , u ( x )) ∀ x ∈∂Ω o , where ν ε Ω i and ν o denote the outward unit normal to ε ∂Ω i 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 .