April 22, 2011  Robert V. Kohn, CIMS

Title: Energy-driven pattern formation: a new frontier for the calculus of variations.
  

  Energy-driven pattern formation examines how energy minimization leads to the formation of defects and microstructure in a variety of physical systems. Examples include the wrinkling of a stretched elastic membrane, the formation of domains in a magnetic material, and the twinning produced by martensitic phase transformation. These systems can be described by ``Landau theories'' -- essentially, nonconvex variational problems regularized by higher-order singular perturbations. I will show in various examples -- some old, some new -- how one can identify the scaling law of the minimum energy; how this sheds light on the underlying patterns; and how the investigation of such issues leads to new types of questions in the calculus of variations.