Publisher Summary This chapter presents material layout problems, which fail to have an optimal design given by a configuration of the two materials. Instead, the behavior of minimizing the sequences of configurations should be studied. The chapter provides a methodology for the recovery of optimal configurations when they exist and identifies minimizing the sequences of configurations otherwise. It also introduces a tractable method for the numerical computation of minimizing the sequences of configurations. These minimizing sequences are associated with materials with graded dielectric properties that may exhibit a fine-scale structure composed of the layers of the two dielectrics. For a dense class of target fields, authors are able to characterize all fine scale structure that can appear in the minimizing sequences of configurations. The examples presented in the chapter illustrate how the electric field can be controlled using functionally graded materials.