Abstract: Nature is full of energy-driven patterns. Some represent local or global minimizers of a suitable free energy. Others are self-organized transients produced by energy-dissipating dynamics. Simulation can demonstrate the adequacy of a model, but it rarely explains "why" a pattern forms. Nonlinear PDE and the calculus of variations can sometimes provide a more global understanding. I'll give four independent lectures on problems of this type, followed by a fifth lecture that's a bit different.
1. BOUNDS ON COARSENING RATES
Some energy-driven systems develop interesting patterns transiently (as they evolve) rather than in steady state (at local minima). An example is the coarsening of a complex initial state under motion by surface diffusion. In this setting (and many others), the "local length scale" increases with time, often with an exponent that can be guessed by dimensional analysis. I'll introduce this phenomenon, then discuss a scheme introduced with F. Otto a few years ago for proving an upper bound on the coarsening rate, focusing on one of the earliest applications: motion by surface diffusion.
2. THE INTERNAL STRUCTURE OF A CROSS-TIE WALL
The cross-tie wall is a particular type of domain wall that forms in soft, thin ferromagnetic films. I'll explain its structure by identifying an associated variational problem, then showing that the pattern we see achieves its minimum. (The main arguments are due to Alouges, Riviere, and Serfaty.)
3. THE SHARP-INTERFACE LIMIT OF ACTION MINIMIZATION
Energy-driven systems typically achieve local not global minima. Thermal fluctuations lead to switching from one local minimum to another. The action functional identifies the rate and most likely pathway of switching. I'll introduce this topic, then discuss the sharp-interface limit of action minimization for the Modica-Mortola functional (work with Otto, Reznikoff, Tonegawa, and Vanden-Eijnden.)
4. THE RELAXATION OF A CRYSTAL SURFACE BELOW THE ROUGHENING TEMPERATURE
The surface energy of a crystal is highly anisotropic: certain planes are strongly preferred. The consequences of such anisotropy for the isoperimetric problem are well-understood: its solution has facets. The consequences for dynamics, however, are much less well-understood. A simple but widely-used model is a fourth-order steepest descent for a convex but non-smooth surface energy. I'll introduce this approach and discuss its main features. Then I'll explain why solutions with "sinusoidal" initial data in one space dimension are in a certain sense asymptotically self-similar (work with Odisharia and Versieux).
5. CLOAKING BY CHANGE OF VARIABLES
We say a region of space is "cloaked" with respect to electromagnetic measurements if its contents -- and even the existence of the cloak -- are inaccessible to such measurements. One recent proposal for achieving cloaking takes advantage of the coordinate-invariance of Maxwell's equations. I'll explain this scheme, including its mathematical basis and its apparent limitations (work with Onofrei, Shen, Vogelius, and Weinstein.)