Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case

For a class of systems of semi-linear elliptic equations, including for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform boundedness of the solutions implies uniform boundedness of their Lipschitz norm as , 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 proofs rest on monotonicity formulae of Alt-Caffarelli-Friedman and Almgren type in the variational setting, and on the Caffarelli-Jerison-Kenig almost monotonicity formula in the symmetric one.