Elementary Soft and Stiff Problems

For a large class of homogenization problems the given periodic matrix A(x) satisfies the usual inequality v₁I ≤ A(x) ≤ v₂I (v₁, v₂ > 0) at all points of ℝm outside certain subsets, called inclusions, where the matrix A(x) is degenerate. Two main types of degeneration are usually considered: if A(x) = 0, then we speak of soft inclusions, or a soft problem; if, on the other hand, the inverse matrix B = 0, then we are dealing with stiff inclusions, or a stiff problem. As a rule, the matrix A(x) is constant outside the inclusions, and therefore we have a kind of two-phase periodic medium.