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We show the local uniqueness of the Cauchy problem for the second order elliptic operators whose coefficients of the principal part are real-valued and continuous with some modulus of continuity. These coefficients are not necessarily lipschitz continuous. The proof is given by drawing the Carleman estimates with a weight attached to the modulus of continuity. §

Local uniqueness in the Cauchy problem for second order elliptic equations with non-Lipschitzian coefficients