Singular perturbation and the energy of folds

We address the singularly perturbed variational problem integral epsilon (- 1)(1 - \del u\(2))(2) + epsilon\del del u\(2) in two space dimensions. We i ntroduce a new scheme for proving lower bounds and show the bounds are asym ptotically sharp for certain domains and boundary conditions. Our results s upport the conjecture, due to Aviles and Giga, that folds are one-dimension al, i.e., del u varies mainly in the direction transverse to the fold. We a lso consider related problems obtained when (1 - \del u\(2))(2) is replaced by (1 - delta(2)U(X)(2) - U-Y(2))(2) or(1 - \del U\(2))(2)gamma.