Mathematics 7710-1.

Part 1. Homogenization

I semester

Introduction:

One dimensional homogenization.
• Canonical form, averaging.
• Examples. effective conductivity, speed of sound, etc.
• Vibrology: constructions under vibrations. Averaged equations. Examples. Vibro-viscosity, granular media.

Introduction to optimal control

• Control theory: variables, controls, functionals. Examples.
• Canonical form and Pontriagin's maximum principle.
• Chattering control and averaging in the optimal systems.

Equations
• Inhomogeneous conducting medium.
• Elasticity equations.
Homogenization technique
• Asymptotic expansion: Effective coefficients.
• Correctors.
• Examples: laminates.
• Homogenization and the boundary conditions.
Effective properties and microstructures.
• Homogenization and Gamma- and G-convergence.
• G-closures. Topological properties.
Exact solutions
• Checker board structure, 2d polycrystal.
• Laminates of high rank.
• Algebra of laminates from contrast materials.

Optimization of the conducting structures
• Wiener bounds or effective properties.
• Minimal energy. Relaxation and homogenization
• Necessary conditions of Weierstrass type.
• Example: an annulus of optimal conductivity.
• Multi-material design.
• Lower weakly semi-continuous functionals. Structure of solutions.
• Examples. How to turn the current away from the field? The thermal lense.

Optimal caverns in an elastic plane

• Optimal cavern in hydrostatic field. Periodic array.
• Optimal cavern in the shear field field. Collective effect.