Abstract
In this paper we consider the so-called prescribed curvature problem approximated by a singularly perturbed double obstacle variational inequality. We extend Ref. 10 with the introduction of the same nonregular potential used for the evolution problem in Ref. 9 and prove an optimal [Formula: see text] error estimate for nondegenerate minimizers (where ε represents the perturbation parameter). Following Ref. 10 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 estimate for discrete minima using the Rannacher–Scott L∞-estimates and under appropriate restrictions on the mesh size ([Formula: see text] with σ>5/2).
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