  The Johnson graph $J(n,m)$ has the $m$--subsets of $\{1,2,\ldots,n\}$ as
vertices and two subsets are adjacent in the graph if they share $m-1$
elements. Shapozenko asked about the isoperimetric function $\mu_{n,m}(k)$ of
Johnson graphs, that is, the cardinality of the smallest boundary of sets with
$k$ vertices in $J(n,m)$ for each $1\le k\le {n\choose m}$. We give an upper
bound for $\mu_{n,m}(k)$ and show that, for each given $k$ such that the
solution to the Shadow Minimization Problem in the Boolean lattice is unique,
and each sufficiently large $n$, the given upper bound is tight. We also show
that the bound is tight for the small values of $k\le m+1$ and for all values
of $k$ when $m=2$.
