# UNIFORM BOUNDS FOR STRONGLY COMPETING SYSTEMS: THE OPTIMAL LIPSCHITZ CASE

created by soave on 29 Jul 2015

[BibTeX]

Accepted Paper

Inserted: 29 jul 2015

Journal: Archive Ration. Mech. Anal.
Year: 2015
Doi: 10.1007/s00205-015-0867-9
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 \emph{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.