| Abstract: |
| We investigate the asymptotic behavior, as $\beta \to +\infty$, of solutions to competition-diffusion system of type
\[
\begin{cases}
\Delta u_{i,\beta} = \beta u_{i,\beta} \prod_{j \neq i} u_{j,\beta}^2 & \text{in }\Omega,\
u_{i,\beta} = \varphi_i \ge 0& \text{on }\partial \Omega,
\end{cases} \quad i=1,2,3,
\]
where $\varphi_i \in W^{1,\infty}(\Omega)$ satisfy the \emph{partial segregation condition}
\[
\varphi_1\,\varphi_2\,\varphi_3 \equiv 0 \quad \text{in $\overline{\Omega}$}.
\]
For $\beta>1$ fixed, a solutions can be obtained as a minimizer of the functional
\[
J_\beta({\bf u},\Omega):= \int_{\Omega} \big( \sum_{i=1}^3 |\nabla u_i|^2 + \beta \prod_{j=1}^3 u_j^2\big)\,dx
\]
on the set of functions in $H^1(\Omega,\R^3)$ with fixed traces on $\partial \Omega$. We prove \emph{a priori} and \emph{uniform in $\beta$} H\older bounds. In the limit, we are lead to minimize the energy
\[
J{\bf u},\Omega):= \int_{\Omega} \sum_{i=1}^3 |\nabla u_i|^2 \,dx
\]
over all partially segregated states:
\[
u_1\,u_2\,u_3 \equiv 0 \quad \text{in $\overline{\Omega}$}
\]
satisfying the given, partially segregated, boundary conditions above. We prove regularity of the free boundary up to a low-dimensional singular set. |
|