Unique continuation for first-order systems with integrable coefficients and applications to elasticity and plasticity

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Let Ω⊂RN be a Lipschitz domain and Γ be a relatively open and non-empty subset of its boundary ∂Ω. We show that the solution to the linear first-order system:(1)∇ζ=Gζ,ζ|Γ=0, vanishes if G∈L1(Ω;R(N×N)×N) and ζ∈W1,1(Ω;RN). In particular, square-integrable solutions ζ of (1) with G∈L1∩L2(Ω;R(N×N)×N) vanish. As a consequence, we prove that:⦀⋅⦀:C∘∞(Ω,Γ;R3)→[0,∞),u↦‖sym(∇uP−1)‖L2(Ω) is a norm if P∈L∞(Ω;R3×3) with CurlP∈Lp(Ω;R3×3), CurlP−1∈Lq(Ω;R3×3) for some p,q>1 with 1/p+1/q=1 as well as detP⩾c+>0. We also give a new and different proof for the so-called ‘infinitesimal rigid displacement lemma’ in curvilinear coordinates: Let Φ∈H1(Ω;R3), Ω⊂R3, satisfy sym(∇Φ⊤∇Ψ)=0 for some Ψ∈W1,∞(Ω;R3)∩H2(Ω;R3) with det∇Ψ⩾c+>0. Then there exists a constant translation vector a∈R3 and a constant skew-symmetric matrix A∈so(3), such that Φ=AΨ+a.