MSI Weekly Bulletin - Week starting Monday 6 March, 2006

The Allen-Cahn equation arises from several applied fields, such as material science and mathematical biology. Recent research has gained deeper understanding of its solutions in the one dimensional case. However, the high dimensional case is much more difficult to understand. It is known that there are solutions with layers, and solutions with spikes. We establish for the high dimensional case the existence of solutions with layers as well as spikes. Our approach relies on sharp estimates of small eigenvalues and some new techniques in the so-called reduction methods.