In this paper, we study a Liouville-type theorem for the Lamé system with rough coefficients in the plane. Let u be a real-valued two-vector in R2 satisfying ∇u ∈ Lp (R2) for some p > 2 and the equation divμ ∇u + (∇u)T + ∇(λ div u) = 0 in R2. When ‖∇μ‖L 2 (R2) is not large, we show that u ≡ constant in R2. As by-products, we prove the weaK unique continuation property and the uniqueness of the Cauchy problem for the Lamé system with small ‖μ‖W 1,2.
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