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.
Categories: math.AP