Internal layers in high-dimensional domains

For a system of semilinear elliptic partial differential equations with a small parameter, defined on a bounded multi-dimensional smooth domain, Rie show the existence of solutions with internal layers. The high-dimensionality of the domain gives rise to quite interesting an outlook in the analysis, dramatically different from that in one-dimensional settings. Our analysis indicates, in a certain situation, an occurrence of an infinite series of bifurcation phenomena accumulating as the small parameter goes to zero. We also present a related free boundary problem with a possible approach to its resolution.