Uniform bounds for strongly competing systems: the optimal Lipschitz case

For a class of systems of semi-linear elliptic equations, including $$-\mathrm{\Delta }{u}_i=f_i(x,u_i)-\beta u_i\sum _{j\mathrm{\ne }i}{a}_{ij}{u}_j^p,i=1,\dots ,k,$$ for $p=2$ (variational-type interaction) or $p=1$ (symmetric-type interaction), we prove that uniform $L^{\mathrm{\infty }}$ 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.