The convex compactification can be used for relaxation (as continuous extension) of various problems in variational calculus and optimization theory.
[1] It may capture both fast oscillations and concentration effects in optimal controls or solutions of variational problems.
In variational calculus, the relaxed problems can serve for modelling of various microstructures arising in modelling Ferroics, i.e. various materials exhibiting e.g. Ferroelasticity (as Shape-memory alloys) or Ferromagnetism.
The first-order optimality conditions for the relaxed problems leads Weierstrass-type maximum principle.
In partial differential equations, relaxation leads to the concept of measure-valued solutions.