# Lipschitz Lectures

## Variational methods for the analysis of patterns

## Mark A. Peletier

University: Technische Universiteit Eindhoven

**Date:** 20 April - 8 May 2009

**Location:** Wegelerstraße 10, University of Bonn, Germany

Schedule:

- Monday April 20, 10-12, lecture in the "kleiner Hörsaal"
- Monday April 20, 12-14, tutorial in the "kleiner Hörsaal"
- Friday April 24, 14-12, lecture in the "Zeichensaal"
- Monday April 27, 10-12, lecture in the "kleiner Hörsaal"
- Monday April 27, 12-14, tutorial in the "kleiner Hörsaal"
- Thursday April 30, 12-14, lecture in the "kleiner Hörsaal"
- Monday May 4, 10-12, lecture in the "kleiner Hörsaal"
- Monday May 4, 12-14, tutorial in the "kleiner Hörsaal"
- Friday May 8, 14-16, lecture in the "Zeichensaal"

### Abstract:

The world abounds with patterns. Some of them are man-made, such as the bricks in a wall and the stitches in a knitted sweater. Others we think of as self-organized, such as the stripes of the zebra or the sandy ripples on the beach. It is the latter category that we find most interesting, which we would like to understand, and hopefully - in some cases - control.

In this lecture series we focus on a class of variational pattern-forming systems. These are systems governed by an energy, in the sense that patterns with lower energy have a higher probability of appearing. Many naturally appearing systems are of this type, and often the 'energy' in the mathematical sense also has a strong physical, chemical, or biological interpretation.

For us the variational structure will be the starting point for an analysis of these systems. We will introduce and use tools which allow us to exploit the variational structure, in order to give us insight in the properties of the solution and how these depend on parameters.

While this course centers around patterns, many of the techniques we introduce and use are not limited to patterns but apply to a wide class of variational problems. Therefore this lecture series may also be interesting for students who simply wish to learn a set of tools for the analysis of variational systems.

This is a rough list of the topics that we will cover: One-dimensional systems (Swift-Hohenberg, extended Fisher-Kolomogorov, strut-on-foundation, Modica-Mortola); existence by minimization, constrained minimization, and mountain-pass techniques; qualitative properties such as generalized monotonicity and 'lego-brick' construction. Higher-dimensional systems: Gamma-convergence as an analysis tool, applications to Modica-Mortola and to droplet structures in Ohta-Kawasaki; lower bounds on energy density. We also comment on the modelling origin of the systems that we study.

### Further information:

Let me first describe the philosophy of the course. The study of patterns has a long history. An important body of pattern analysis comes from bifurcation theory; the books by Hoyle (Pattern Formation, 2006) and Peletier-Troy (Spatial Patterns in Higher-Order Models in Physics and Mechanics, 2001) give an excellent overview of this type of approach (although the book by my father and Bill Troy also uses variational arguments).

I am myself more interested in the variational theory of patterns, for two reasons. The first is that I have a general preference for variational problems; I like the variational structure, the intuition that it gives me, the connections with physics that often accompany such a structure, and the tools that variational systems allow me to use. The second is that the variational structure can allow for stronger statements, such as global characterizations instead of local ones, characterization of stability properties, strong limit characterizations such as by Gamma-convergence, and many more.

For this lecture series I chose for variational tools in pattern formation partly because of this personal preference, but also because it allows us to focus on a specific type of question - understanding pattern techniques that apply to a wide range of variational problems. The focus for this series of lectures will therefore be on variational tools that are well-suited for the analysis of patterns, and which will also serve you well in different problems.

In case you're interested in introductory reading material, here is a short list.

- On the general topic of patterns, take a look at the introduction of the Peletier-Troy book (I hope to provide a pdf file). I like its description of model equations and the role they play. The introduction doesn't contain many examples, however.
- If you want to see a wide variety of general examples of pattern-forming systems, have a look at the reading material of the course of Roman Grigoriev. It's a physicists' exposition, long on phenomena and short on rigour, but it shows very nicely the broad range of pattern formation in nature and science. Keep in mind that that the rigorous analysis that we will be doing is much less spectacular, although it is - at least in my eyes - more rewarding :-)

As for required background in mathematical tools:

- Most of the appendices of Evans (Partial Differential Equations, 2002) will be used at one moment or the other (maybe with the exception of Fredholm theory and Bochner integrals). Particularly important tools are the theory of Sobolev spaces W^{k,p}, and their properties, such as embedding, trace, and compactness theorems (in Evans these are found in chapter 5: in particular, the results of sections 5.1-5.3, 5.5-5.6, and 5.8.1 will be used).
- We will be using weak solution concepts, mostly for elliptic equations; Sections 6.1, 6.2.1, and 6.2.2 of Evans nicely describe the relevant concepts.
- All of the weak solutions will be obtained as stationary points of a functional; Sections 8.1.2 and 8.2.1-8.2.3 cover this approach.

I'm assuming at the moment that most of the students will know most of this material; but to be completely sure, I want to spend some time on the first day (April 20) on the background knowledge, to make sure that you and I are on the same wavelength.

- Switching from prerequisites to the course content itself, some of the tools that we will introduce during the lectures also appear in Chapter 8 of Evans. We will be using constrained minimization (Section 8.4) and the mountain-pass theorem (8.5.1).