Uniform Bounds for Strongly Competing Systems: The Optimal Lipschitz Case

For a class of systems of semi-linear elliptic equations, including (Formula presented.), for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform L^∞ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β→+∞, lthat 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.