Abstract: The equilibria of many physical systems are determined by minimization of an appropriate free energy. If the free energy is non-convex, the ground states can display various multiple-scale behaviors including singularities, defects and micro-structure. The following questions are of practical importance in such systems:
(a) How does one analyze/predict/control multiple scale behaviors for a specific system?
(b) Can we elucidate principles that help us understand multiple-scale phenomena in a general context?
I will illustrate these issues using two physically relevant examples -- Thin elastic sheets and the convection roll patterns. I will review recent results from the rigorous analysis of these problems. I will also discuss the implications of these results to the question of what (if any) general principles govern multiple scale behaviors in non-convex variational problems.