Sharp lower bounds for vector Allen-Cahn energy and qualitative properties of minimizes under no symmetry hypotheses

Abstract

We study vector minimizers u of the Allen-Cahn functional with potentials possessing N global minima defined on bounded domains, with certain geometrical features and Dirichlet conditions on the boundary. We derive a sharp lower bound for the energy (as ε{\rightarrow} 0) with the additional feature that it involves half of the gradient and part of the domain. Based on this we derive very precise (in ε) pointwise estimates up to the boundary for uε. Depending on the geometry of the domain uε exhibits either boundary layers or internal layers. We do not impose symmetry hypotheses.