Lemma 5.2 [0324]

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Lemma 5.2

(The $C^{0}$ estimate.) There exists a constant $C$ depending only on $J$ , such that for all $\epsilon<\epsilon_{0}$ , ( $\epsilon_{0}$ as in Theorem 4.5)

\|u_{\epsilon}\|_{\infty}\leq C\epsilon^{-5}e^{-D_{2}/\epsilon}.