Uniform bounds for strongly competing systems: the optimal Lipschitz case

For a class of systems of semi-linear elliptic equations, including $-Δu_i=f_i(x,u_i) - βu_i\sum_{j\neq i}a_{ij}u_j^p,\qquad i=1,\dots,k,$ 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 $β\to +\infty$, 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.

No reviews yet.

No comments yet.