# Quasiconvexity and its applications

## Abstracts for the talks

### Quasiconformal mappings, G-closures and Young measures

**Kari Astala** *(Helsinki)*

Friday, November 15, 2002

In two dimensions, in particular, the quasiconformal mappings provide promising tools for many open questions connected to understanding the notion of quasiconvexity and related topics. Actually, in recent years we have seen a reciprocal exchange of ideas between these two fields.

In the talk we will give an overview of these interactions, in the case of two dimensions.

### Quasiconvex hulls and minimization in the calculus of variations

**Bernard Dacorogna** *(EPFL Lausanne)*

Thursday, November 14, 2002

I will discuss several results on the computation of rank one convex hulls for problems involving singular
values. I will then show how to apply these results to some non quasiconvex minimization problems of
the calculus of variations.

### Quasiconvexity and polyconvexity in elastodynamics

**Constantine Dafermos** *(Brown)*

Saturday, November 16, 2002

The lecture will attempt to demonstrate that the notions of quasiconvexity and polyconvexity, whose importance in elastostatics is well documented, are also relevant in elasodynamics.

### Tutorial on quasiconvexity and regularity of minimizers

**Lawrence Craig Evans** *(Berkeley)*

Friday, November 15, 2002

In this expository lecture I will explain how to show partial regularity of minimizing mappings for both convex and quasiconvex energy integrands. The technique is to linearize by ``blowing up'' and the basic issue is establishing compactness of a rescaled sequence of functions.

I will discuss also how to modify these ideas in a case that we have a determinant constraint.

### Cauchy-Born rule and crystallization in 2D mass-spring lattices

**Gero Friesecke** *(Warwick)*

Saturday, November 16, 2002

The Cauchy-Born (CB) rule postulates that when a monatomic crystal is subjected to a small linear displacement of its boundary, then all atoms will follow this displacement. In the absence of previous mathematical results, we study the validity of this rule in the model case of a 2D cubic lattice interacting via harmonic springs between nearest and diagonal neighbours.

Establishing validity of the CB rule is a variant of the celebrated ``crystallization problem''. Another variant, obtained by replacing linear displacement boundary conditions by zero applied forces, will also be discussed in my lecture.

We find that for favourable values of the spring constants and spring equilibrium lengths the CB rule is a theorem, and that for unfavourable values the rule is incorrect. Moreover in the latter case the overestimation of the lattice energy per unit volume by the CB rule cannot be cured by quasiconvexification (and not even by convexification) of the CB energy.

Roughly speaking, validity of the CB rule can be viewed as a lattice analogon of quasiconvexity, and our proof of CB goes via verifying an appropriate analogon of polyconvexity, which we call lattice polyconvexity.

(In case of zero applied forces, the tool of lattice polyconvexity has to be replaced by a recent geometric rigidity theorem due to Friesecke, James and Müller.)

Reference: G.Friesecke, F.Theil, J.Nonl.Sci. 12 No. 5 2002 445-478

### On the total variation of the Jacobian

**Nicola Fusco** *(Napoli)*

Thursday, November 14, 2002

We shall discuss a representation formula for the total variation of the determinant of W^{1,p} maps from R^{n} into a R^{n}, where n-1 < p < n, having a singularity at a point.

### What's new for Jacobian determinants and rank-one connections

**Tadeusz Iwaniec** *(Syracuse (NY))*

Saturday, November 16, 2002

Geometric analysis of Sobolev mappings (weakly differentiable deformations) of Euclidean domains or Riemannian n-manifolds will continue to enhance mathematical insights into several applied fields: nonlinear elasticity, material science, crystals, and so forth. In this challenge, there is an important place for the Jacobian determinant of the gradient matrix, its sub-determinants and other null-Lagrangians. We shall emphasize the fundamental role of the Orlicz-Sobolev spaces to capture fine properties of those differential expressions. This brings us closer to PDEs (Hodge theory and very weak solutions) and modern techniques of harmonic analysis (BMO and Hardy spaces, spherical maximal operator, nonlinear commutators and interpolation). Among the novelties we shall give various estimates of the Jacobian by means of the cofactors of the gradient matrix.

Functional analytic properties of subdeterminants have something to teach us about rank-one connections between Lipschitz mappings at the interface, the central issue in the geometry of microstructure and crystals. Unexpectedly, the desired rank-one connections may not be found in the polyconvex hulls of the gradients, leaving some concerns about the mathematical models of microstructure.

The nagging problem concerning Morrey's conjecture for 2x2 -matrices seems to lay beyond the power of the existing methods, but numerous links to other outstanding questions in analysis are helping drive the subject. Among them we shall discuss the fundamental L^{p}-inequality of the Jacobian, as a base for the analytic theory of quasiconformal mappings.

We shall briefly mention recently discovered phenomena about Jacobian of mappings between Riemannian manifolds.

### Quasiconvexity and hysteresis

**Richard James** *(Minnesota)*

Thursday, November 14, 2002

Hysteresis is a universal feature of first order phase transformations. During observation of the transformation, this may present itself as different transformation temperatures on heating and cooling, or different transformation stresses on loading and unloading. One way to approach this mathematically is through a study of local minimizers. A natural concept for local minimizer emerges from the constraint of geometric compatibility, and has links to the concept of quasiconvexity (joint work with J. M. Ball). This concept seems to explain, at least qualitatively, the hysteresis observed in some biaxial tests the author did with C. Chu. However, it clearly does not explain the hysteresis observed in many other experiments, for example, the simple experiment of measuring transformation temperatures upon heating and cooling. By surveying the hysteresis in lots of systems, we arrive at a related but different concept. At this time of the writing of this abstract, it is not clear whether this new concept also has links to quasiconvexity, but in any case it relates in some way to the idea that, if a material has certain special distortions, then the phases fit together unusually well.

### Solving partial differential inclusions

**Bernd Kircheim** *(MPI Leipzig)*

Saturday, November 16, 2002

We are interested in mappings whose gradient takes values in a given (finite) set of matrices.

Existence results rely on constructions via laminations and hence rank-one convexity, where as non-existence (or regularity) uses properties of general gradients and is more related to quasiconvexity and, in particular, quasiconformality.

It turns out that these two approaches complement each other surprisingly well when we consider either sets without rank-one connections or the interface problem.

### A construction of quasiconvex functions

**Jan Kristensen** *(Heriot-Watt)*

Thursday, November 14, 2002

It is well-known that polyconvexity implies quasiconvexity, and that the converse is false. In fact, the class of quasiconvex functions on two by two matrices R^{2x2} is in a sense much larger than the corresponding class of polyconvex functions. Despite this there are not many explicit examples of quasiconvex, non-polyconvex functions on R^{2x2} in the literature. Besides this there are interesting problems that can be restated as problems about quasiconvexity of associated functions, and it is often easy to show that these functions are rank-one convex, but that they cannot be polyconvex.

In this talk, which is based on joint work with Tadeusz Iwaniec, we present a construction which yields explicit examples of quasiconvex, non-polyconvex functions. The construction is based on the observation that given any suitably rank-one convex function *R* and any strongly quasiconvex function *P* the function *R+tP* is quasiconvex for sufficiently large numbers *t*. We regard the function *R* as the function that we ideally would like to show is quasiconvex and regard *tP* as a perturbation. The method is illustrated on two well-known families of functions, and it involves local polyconvexity.

### Quasiconvexity, rank-one convexity, and composites

**Graeme Milton** *(Utah)*

Friday, November 15, 2002

In this lecture I will discuss some connections between quasiconvexity, rank-one convexity, and the problem of determining the range which the effective tensor of a linear composite can take. It has been known since the work of Kohn that the problem of bounding the effective tensor (such as the effective elasticity tensor or effective conductivity tensor) reduces to a quasiconvexification problem. The converse is true also: subject to some minor technical points, a given quasiconvexification problem can reinterpreted as a problem of finding a bound on an effective tensor.

One interesting topic, initiated by Grabovsky, is the theory of exact relations. An exact relation holds if the set of effective tensors lies within a manifold. Grabovsky found an algebraic condition which ensures an exact relation holds for laminated composites. Together with Sage we found a sufficient algebraic condition for an exact relation to hold for all composites. Curiously, it appears there is a gap between these conditions that is in many ways parallel to the gap between quasiconvexity and rank-one convexity. However in all known examples, if one algebraic condition is satisfied then the other is too.

It is well known that rank-one convexity is equivalent to quasiconvexity if and only if the Young's measure of any gradient can be reproduced by a laminate field. Briane, Nesi and myself have recently probed an analogous question for three dimensional conducting composites. Specifically we find that the determinant of the field maintains its sign in any laminate microstructure but can change its sign in an appropriately chosen microstructure of interlinked chains.

### The mechanics of viral DNA packaging

**Michael Ortiz** *(Caltech)*

Friday, November 15, 2002

DNA is packaged by a portal motors within viral capsids of size comparable to the persistence length of the DNA, and thus the packaging carries a steep cost in terms of elastic energy. The scale of the process is equivalent to packing 20 Km of thread in a tennis ball (Alberts et al., Essential Cell Biology, 1997). A number of novel experimental procedures provide data and insight into the packaging process. For instance, Smith et al.~(Nature, 413 (2001) 748) have measured the force required to package the DNA into a Phi29 bacteriophage as a function of the fraction of genome packed. Cryo-electron microscopy experiments have revealed the concentric arrangements of DNA within viral capsids (Ceterelli et al., Cell, 91 (1997) 271; Olson et al., Virology, 279 (2001) 385).

The objective of the present work is to formulate a continuum model of viral DNA packaging. The conformations of the DNA are described in terms of a director field whose point values give the local direction and density of the DNA. The continuity of the DNA strand requires the director field to be divergence-free and tangent to the capsid wall. The operative principle which determines the DNA conformation is assumed to be energy minimization. The energy of the DNA is defined as a functional of the director field which accounts for the bending and torsion of the DNA strand, as well as for charge and hydration forces. The central problem concerns the determination of the energy-minimizing DNA conformations as a function of fraction of genome packed. A baseline conformation is furnished by a simple inverse-spool geometry. We show that the energy of the inverse-spool conformation may be lowered by a construction consisting of the packing of toroidal coils. The predictions of the theory are compared against the available experimental evidence.

### Universal deformations

**Vladimir Sverak** *(Minnesota)*

Thursday, November 14, 2002

We discuss classes of mappings which can be unseful for testing a certain type of conjectures in the calculus of
variations and regularity theory.

### Multi-linear operators with modulation symmetry

**Christoph Thiele** *(UCLA)*

Saturday, November 16, 2002

This will be a basic survey of some recent (past 7 years) developments in singular integral theory wich are specific to the multi-linear setting, in which modulation symmetry becomes a possibility. Large part of the work presented is joint with M. Lacey, or C. Muscalu and T. Tao.

### On second order multiple integrals in the calculus of variations

**Neil Trudinger** *(ANU Canberra)*

Friday, November 15, 2002

In this talk we consider variational problems for multiple integrals with integrands containing second order partial derivatives , with fourth order Euler-Lagrange equations. A particular example is the affine area functional , which is invariant under uni-modular affine transformations of the ambient space. We shall discuss the existence and uniqueness of maximisers in appropriate weak formulations, the solvability of boundary value problems, issues of regularity and Bernstein-Liouville type results.

### Subspaces without rank-one matrices and quasiconvexity

**Kewei Zhang** *(Sussex)*

Friday, November 15, 2002

I give a survey on linear subspaces of real matrices without rank-one connections (SNR1) and their applications to quasiconvexity. I will discuss (i) a simple case of a two point set without rank-one connections, recalling the compactness result of Ball-James and the quasiconvex envelope of Kohn and Pipkin; (ii) some direct applications and results motivated from (i); (iii) some algebraic, topological and analytic properties of SNR1; (iv) applications to the quasiconvex hull, the unstable solution for a two-point problem and the separation of gradient Young measures.

## Date and Location

**November 14 - 16, 2002**

Princeton University

Princeton

USA

## Scientific Organizers

**John Ball**

University of Oxford

Mathematical Institute

**Weinan E**

Princeton University

Department of Mathematics

**Robert Kohn**

New York University

Department of Mathematics, Courant Institute

**Stefan Müller**

Max Planck Institute for Mathematics in the Sciences