Explore a lecture on elliptic singular perturbed systems and their singular limit to phase segregating systems. Delve into the study of existence, uniqueness, and asymptotic behavior as the interaction rate approaches infinity. Examine the resulting free boundary problem where components cannot coexist simultaneously. Discover a novel method providing explicit solutions for specific parameter choices. Gain insights into numerical simulations of the asymptotic problem. Learn about this research conducted within the Hausdorff Trimester Program on Evolution of Interfaces, presented in a 49-minute talk at the Hausdorff Center for Mathematics.