| Contents |
| We present some new optimal regularity results regarding the family of systems
\[
- \Delta u_{i,\beta} = f_{i,\beta}(x,u_{1,\beta}, \dots, u_{k,\beta} ) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^p
\]
in the case $p=1$ and $p=2$. For such systems, of interest in the study of phase-separation and pattern-formation phenomena, we show that under very mild assumptions on the non-linear terms $f_{i,\beta}$, uniform $L^\infty$ bounds imply corresponding uniform Lipschitz bounds. These results extend the regularity theory available in the literature to the optimal case. |
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