Department of Mathematics, National Tsing Hua University

Abstract：
We study a variational problem motivated by models of species population density in a nonhomogeneous environment. Our focus is on the structure of the saturated region, the set where the population reaches its maximal density, from a free-boundary perspective. The regularity of this region is established through an analysis of the blow-up profiles associated with the Euler-Lagrange equation. Numerical simulations are presented to illustrate the theoretical results and to provide further insight into spatial saturation patterns in population models. We also highlight two open problems: one concerning the quasiconcavity of minimizers, and the other, the classical Pompeiu problem (including some of our observations). This talk is based on joint work with Masato Kimura and Hiroshi Ohtsuka. The project originated during my visits to the co-authors at Kanazawa University in March and August 2025, both generously supported by the university.