A direct algorithm for constrained variational problems in several dimensions

We introduce a heuristic, practical procedure to take into account, in an easy-to-implement way, point-wise constraints in a variational problem in several dimensions. In addition to showing a convergence result under suitable assumptions, we emphasize the flexibility of the method by using it for an optimal design problem either in conductivity or elasticity, where an optimal mixture of two materials is to be found in a given design domain, under pointwise constraints. We illustrate the performance of the algorithm with some simple test examples.