Cliques in graphs with bounded minimum degree

Let $k_r(n,\delta)$ be the minimum number of $r$-cliques in graphs with $n$ vertices and minimum degree $\delta$. We evaluate $k_r(n,\delta)$ for $\delta \leq 4n/5$ and some other cases. Moreover, we give a construction, which we conjecture to give all extremal graphs (subject to certain conditions on $n$, $\delta$ and $r$).