Optimal interface error estimates for a discrete double obstacle approximation to the prescribed curvature problem

In this paper we consider the so called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend [Pao97] with the introduction of the same nonregular potential used for the evolution problem [NPV94c] and we prove an optimal O(e2) error estimate for nondegenerate minimizers (where e stands for the perturbation parameter). Following [Pao97] the result relies on the construction of precise barriers suggested by formal asymptotics combined with the use of the maximum principle. Key ingredients are the construction of a sub(super)solution containing appropriate shape corrections and the use of a modified distance function based on the principal eigenfunction of the second variation of the prescribed curvature functional. This analysis is next extended to a piecewise linear finite element discretization of the elliptic PDE of bistable type to prove the same error extimate for discrete minima using the Rannacher-Scott L\infty-estimates and under appropriated restrictions on the mesh size (h2=O(es) with s > 5/2).