Uniform bounds for strongly competing systems: the optimal Lipschitz case

TL;DR

This paper establishes that for certain strongly competing elliptic systems, uniform boundedness of solutions implies uniform Lipschitz bounds as competition intensifies, extending optimal regularity results.

Contribution

It proves uniform Lipschitz bounds for solutions of strongly competing elliptic systems in the optimal case, using advanced monotonicity formulae.

Findings

Uniform boundedness implies uniform Lipschitz bounds as competition parameter grows.

Extends regularity results to the optimal case for these systems.

Utilizes monotonicity formulas of Alt-Caffarelli-Friedman, Almgren, and Caffarelli-Jerison-Kenig.

Abstract

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 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.