The formation of codimension-one interfaces for multi-well gradient-driven problems  is well-known and established in the scalar case,   where the equation  is often referred  to as the Allen-Cahn equation. The vectorial case  in contrast is quite open.  This lack of results and insight  is to a large extent  related  to the absence  of known  monotonicity formula.  After a presentation of the thory in the scalar case,  I will  focus on the  elliptic case in two dimensions, and presents  some results  which   extend to the vectorial case in two dimensions  most of the results obtained for the scalar case. I will also  emphasize some specific features of the vectorial case.

10.31.pdf