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We study the insulated conductivity problem with inclusions embedded in a bounded domain in Rn.When the distance of inclusions,denoted by ε,goes to 0,the gradient of solutions may blow up.When two inclusions are strictly convex,it was known that an upper bound of the blow-up rate is of order ε-1/2 for n = 2,and is of order ε-1/2+β for some β＞0 when dimension n≥3.In this paper,we generalize the above results for insulators with flatter boundaries near touching points.