Singular perturbation and the energy of folds

We address the singularly perturbed variational problem ∫ ∈^-1(1-|∇u|²)²+ ∈|∇∇u|² in two space dimensions. We introduce a new scheme for proving lower bounds and show the bounds are asymptotically sharp for certain domains and boundary conditions. Our results support the conjecture, due to Aviles and Giga, that folds are one-dimensional, i.e., ∇u varies mainly in the direction transverse to the fold. We also consider related problems obtained when (1 - |∇u|²)² is replaced by (1 - δ²u²_x - u²_y)² or (1 - |∇u|²)^2γ.