The Stokes Paradox in Inhomogeneous Elastostatics
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The Stokes Paradox in Inhomogeneous Elastostatics Adele Ferone1 · Remigio Russo1 · Alfonsina Tartaglione1
Received: 24 September 2018 © Springer Nature B.V. 2020
Abstract We prove that the displacement problem of inhomogeneous elastostatics in a two– dimensional exterior Lipschitz domain has a unique solution with finite Dirichlet integral u, vanishing uniformly at infinity if and only if the boundary datum satisfies a suitable compatibility condition (Stokes paradox). Moreover, we prove that it is unique under the sharp condition u = o(log r) and decays uniformly at infinity with a rate depending on the elasticities. In particular, if these last ones tend to a homogeneous state at large distance, then u = O(r −α ), for every α < 1. Keywords Inhomogeneous elasticity · Two–dimensional exterior domains · Existence and uniqueness theorems · Stokes paradox Mathematics Subject Classification (2010) Primary 74B05 · 35J47 · 35J57 · Secondary 45A05
1 Introduction Let Ω be an exterior Lipschitz domain of R2 . The displacement problem of plane elastostatics in exterior domains is to find a solution to the equations div C[∇u] = 0 in Ω, u = uˆ on ∂Ω, lim u(x) = 0,
r→+∞
B A. Tartaglione
[email protected] A. Ferone [email protected] R. Russo [email protected]
1
Department of Mathematics and Physics, University of Campania “Luigi Vanvitelli”, Caserta, Italy
(1)
A. Ferone et al.
where u is the (unknown) displacement field, uˆ is an (assigned) boundary displacement, C ≡ [Cij hk ] is the (assigned) elasticity tensor, i.e., a map from Ω × Lin → Sym, linear on Sym and vanishing in Ω × Skw. We shall assume C to be symmetric, i.e., Cij hk = Chkij and positive definite, i.e., μ0 |E|2 ≤ E · C[E] ≤ μe |E|2 ,
∀ E ∈ Sym,
a.e. in Ω.
(2)
By appealing to the principle of virtual work and taking into account that ϕ ∈ C0∞ (Ω) is 1,q an admissible (or virtual) displacement, we say that u ∈ Wloc (Ω) is a weak solution (variational solution for q = 2) to (1)1 provided ∇ϕ · C[∇u] = 0, ∀ϕ ∈ C0∞ (Ω). Ω
A weak solution to (1) is a weak solution to (1)1 which satisfies the boundary condition in the sense of the trace in Sobolev’s spaces and tends to zero at infinity in a generalized sense. 1,q If u ∈ Wloc (Ω) is a weak solution to (1) the traction field on the boundary s(u) = C[∇u]n exists as a well defined field of W −1/q,q (∂Ω) and for q = 2 the following generalized work and energy relation [9] holds ∇u · C[∇u] = u · s(u) + u · s(u), ΩR
∂Ω
∂SR
for every R such that SR ⊃ Ω , where with abuse of notation by Σ u · s(u) we mean the value of the functional s(u) ∈ W −1/2,2 (Σ) at u ∈ W 1/2,2 (Σ) and n is the unit outward (with respect to Ω) normal to ∂Ω. It will be clear from the context when we shall refer to an ordinary integral or to a functional. It is a routine to show that under assumption (2), (1)1,2 has a unique solution u ∈ D 1,2 (Ω), we shall call D–solution (for the notation see at the end of this section). Moreover, it exhibits more regularity provided C, ∂Ω and uˆ are more re
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