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

Solutions of the so-called prescribed curvature problem
$\min _{A\subseteq \Omega } \mathcal {P}_ \Omega (A) - \int _A g(x)$, $g$ being the curvature field, are approximated via a singularly perturbed elliptic PDE of bistable type. For nondegenerate relative minimizers $A \subset \subset \Omega$ we prove an $\mathcal {O}( \epsilon ^2 |\log \epsilon |^2)$ 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.