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.