A quasi-optimal error estimate for a discrete singularly perturbed approximation to the prescribed curvature problem

Solutions of the so-called prescribed curvature problem min(A subset of or equal to Omega) P-Omega(A) - integral(A)g(x), g being the curvature field, are approximated via a singularly perturbed elliptic PDE of bistable type. For nondegenerate relative minimizers A subset of subset of Omega we prove an O(epsilon(2) \log epsilon\(2)) error estimate (where epsilon stands for the perturbation parameter), and show that this estimate is quasi-optimal. The proof is based on the construction of accurate barriers suggested by formal asymptotics This analysis is next extended to a finite element discretization of the PDE to prove the same error estimate for discrete minima.