Variational principles with integrands defined through a minimum

Abstract

We revisit a systematic approach for the computation and analysis of the convex hull of non-convex integrands defined through the minimum of convex densities. It consists in reformulating the non-convex variational problem as a double minimization and exploiting appropriately the nature of optimality of the inner minimization. This requires gradient Young measures in the vector case, even if the initial problem was scalar, as the full problem is recast through the computation of a certain quasiconvexification. We illustrate this strategy by looking at two typical non-convex scalar problems. We hope to address vector problems in the near future.