Uniform bounds for strongly competing systems: the optimal Lipschitz case

Abstract

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 $\beta \to +\infty$, 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.

to appear on Archive for Rational Mechanics and Analysis

References in corpus (1)

• Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation

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