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 {\epsilon}{\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 {\epsilon}) pointwise estimates up to the boundary for u{\epsilon}. Depending on the geometry of the domain u{\epsilon} exhibits either boundary layers or internal layers. We do not impose symmetry hypotheses.