= Solution
For fixed $k$, <independence> and the <product large-deviation principle> give the vector $(Y_n(1),\ldots,Y_n(k))$ the <good rate function> $\sum_{j=1}^k I(y_j)$. Since the minimum is a <continuous map>, the <contraction principle for large deviations> gives
$$
J(m)=\inf_{\min_jy_j=m}\sum_{j=1}^kI(y_j).
$$
In every vector in this fibre, at least one coordinate equals $m$ and all others are at least $m$. The <large-deviation rate of a minimum of independent copies> therefore reduces the infimum to
$$
J(m)=I(m)+(k-1)\inf_{y\ge m}I(y).
$$
If $0<m<\mu=\lambda^{-1}$, the other coordinates can equal $\mu$ and cost zero. If $m\ge\mu$, the <rate function> is increasing on $[\mu,\infty)$, so every coordinate costs at least $I(m)$ and equality is achieved when all coordinates equal $m$. For $m\le0$ the cost is infinite. Thus
$$
\boxed{J(m)=\begin{cases}I(m),&m<\lambda^{-1},\\kI(m),&m\ge\lambda^{-1}.\end{cases}}
$$
At the threshold both expressions are zero. Goodness follows from the <contraction principle for large deviations>. A small minimum requires only one atypically small mean; a large minimum requires all $k$ means to be atypically large, which explains the two costs.
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