  For a class of systems of semi-linear elliptic equations, including \[
  -\Delta u_i=f_i(x,u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^p,\qquad
i=1,\dots,k, \] for $p=2$ (variational-type interaction) or $p = 1$
(symmetric-type interaction), we prove that uniform $L^\infty$ boundedness of
the solutions implies uniform boundedness of their Lipschitz norm as $\beta \to
+\infty$, that is, in the limit of strong competition. This extends known
quasi-optimal regularity results and covers the optimal case for this class of
problems. The proof rests on monotonicity formulae of Alt-Caffarelli-Friedman
and Almgren type in the variational setting and Caffarelli-Jerison-Kenig in the
symmetric one.
