# Geometric Analysis, Elasticity and PDE

*23 - 27 Jun 2008*

*Heriot Watt University, Edinburgh*

### Organiser

Name | Institution |
---|---|

Bona, Jerry | University of Illinois at Chicago |

Carr, Jack | Heriot-Watt University |

James, Richard | University of Minnesota |

Marsden, Jerry | California Institute of Technology |

Murat, François | Université Pierre et Marie Curie (Paris VI) |

Slemrod, Marshall | University of Wisconsin |

**Scientific Organiser**

Jack Carr**Scientific Committee**

Professor Jerry Bona, Co-Chair (University of Illinois)

Professor Jack Carr, Heriot-Watt University

Professor Dick James (University of Minnesota)

Professor Jerry Marsden, Co-Chair (California Institute of Technology)

Professor Stefan Müller (Max Planck Institute for Mathematics, Leipzig)

Dr François Murat (Université Pierre et Marie Curie, Paris VI)

Professor Marshall Slemrod (University of Wisconsin)**Local Organising Committee**

Professor Tony Carbery (University of Edinburgh)

Professor Jack Carr (Heriot-Watt University)

Professor Chris Eilbeck (Heriot-Watt University)

Professor Robin Knops (Heriot-Watt University)

Professor John Toland (University of Bath)

2008 sees the 60th birthday of Sir John Ball, FRS, FRSE, and we will be celebrating this occasion during the workshop. Lecture notes and photographs from the workshop are now available on a page maintained by Professor Eilbeck.

Nonlinear Partial Differential Equations (PDE) are of fundamental importance in mathematics and although major progress in their study has recently been achieved, many profound challenges remain. Developments are motivated not only by advances in pure and applied mathematics but also by new issues in the physical and biological sciences, mathematical finance and risk theory and many other applied disciplines.

Analytical, dynamical systems, as well as geometric methods continue to play a critical role in both fundamental investigations as well as numerical methods in solid and fluid mechanics, including complex materials, such as material microstructures and liquid crystals. It is clear that this subject is in a period of exciting growth and development. The principal aim of the workshop is to promote international exchange of recent developments in the analysis and geometry of PDE and their applications.

Specific objectives include:

- mapping new frontiers of research
- building bridges between the most recent developments in the general mathematical theory and the areas of applications that have developed most actively in the last few years,
- identifying new applications of PDE

Specific themes of the workshop include:

- Nonlinear elasticity and liquid crystals
- Material science; microstructure, minimizer in the presence of symmetry, relation with experiment
- Calculus of variations
- Systems of conservation laws
- Transport equations
- Biomaterials Dynamical systems methods and the PDEs of mechanics

**Supporting Institutions** **EPSRC****Maxwell Institute for Mathematical Sciences** (Centre for Analysis and Nonlinear PDE)**University of Oxford****LMS****Edinburgh Mathematical Society**

### Arrangements

**Participation **You may apply to attend this meeting by completing the application form which is available at the top of this page. (Please note that the application form will close on 23 May, or when we have reached the maximum number for the meeting, 150.) You will then receive an e-mail confirming your place.

*Speakers and committee members will receive a separate invitation via e-mail.*

If you **do not** receive an e-mail please contact Morag Burton.

**Venue **

The Workshop will be held at Heriot-Watt University. The campus is on the western edge of the city, near the airport, with excellent public transport to and from Edinburgh city centre. Click **here** for instructions on how to reach the university by various forms of transport.

**Registration**

There will be a combined Registration and Welcome Buffet on Sunday 22 June, at Heriot-Watt University. Those who cannot register on Sunday evening may do so on Monday 23 June prior to the start of the lectures or during the first coffee break. There will be a Registration Fee of 60.00 GBP. For those of you who want to pay by credit card, the credit card form can be found here. You may post or fax (+44 (0)131 220 1053) this to us in advance or bring it with you, already filled in, to registration. Please note that there is a 1.75% charge for use of credit cards

**Poster Sessions**

Contributed presentations in poster format are invited in all areas consistent with the conference themes. If you wish to submit a poster please submit a title and abstract (maximum 75 words) to J. Carr (at) ma.hw.ac.uk and a prompt decision will be made on its acceptability. Information on the format of the poster sessions will be available in March 2008.

**Audio/Visual Facilities**

A data projector, overhead projector, PC, laptopconnection and blackboards will be available.

**Accommodation and Facilities**

Single study-bedrooms are available at Heriot-Watt University. To book contact the Edinburgh Conference Centre directly on +44 (0)131 451 3669 or e-mail info@edinburgh-conference.com .*Invited participants should refer to their personal invitation letter for accommodation arrangements.*

All rooms have a private shower and toilet and a telephone. The telephone will access incoming calls. You need to buy a telephone card from Reception in order to make outgoing calls. Each room also has an internet point should you wish to bring your laptop, although there is a charge for this service. Free email and internet access will be arranged in the central computer labs, the Library and to the university wireless network.

**Meals and Refreshments**

Breakfast will be provided for Heriot-Watt University residents for the period of their stay in the Middle Floor Dining Room.

Refreshments will be available during workshop breaks.

There will be a buffet supper (free of charge to participants) on the evening of Sunday 22 June between 19.00 and 20.30. Participants may register for the workshop during this buffet.

The Workshop Dinner will be on Thursday 26 June. Non-invitees are welcome to come to this on payment of £30.00, which will cover the dinner and wine. Payment for the dinner may be added to the amount being paid for the Registration Fee (see above).

### Programme

**Provisional Timetable**

**Monday 23 June**

09:00-09:30 | John Toland: Welcome |

Chair: | Jack Carr |

09:30-10:30 | Constantine Dafermos (Brown University) Elastodynamics and John Ball |

10:30-11:00 | Refreshments |

11:00-12:00 | Stuart Antman (University of Maryland) Asymptotics of small inertias and large times in nonlinear continuum mechanics |

12:00-13:00 | Carme Calderer (University of Minnesota)Nonlinear elasticity and liquid crystals in biological and biomedical application |

13:00-14:30 | Lunch |

Chair: | Tony Carbery |

14:30-15:30 | Maria J. Esteban (CEREMADE) Critical magnetic field for the Dirac-Coulomb model |

15:30-16:00 | Refreshments |

16:00-17:00 | Vladimir Sverak (University of Minnesota) Liouville theorems for the Navier-Stokes equations |

17:00-18:00 | Poster Viewing |

**Tuesday 24 June**

Chair: | Robin Knops |

09:30-10:30 | Stefan Müller (Max Planck Institute for Mathematics, Leipzig) Some thoughts about crystals and the Cauchy-Born rule at positive temperature |

10:30-11:00 | Refreshments |

11:00-12:00 | Sergio Conti (Universität Bonn) Patterns in thin elastic sheets |

12:00-13:00 | Philippe Ciarlet (City University of Hong Kong) Intrinsic elasticity |

13:00-14:30 | Lunch and Poster Viewing |

Chair: | Dick James |

14:30-15:30 | Nick Schryvers (University of Antwerp) Nano- and microstructures charaterized by transmission electron microscopy |

15:30-16:00 | Refreshments |

16:00-17:00 | Kaushik Bhattacharya (California Institute of Technology) Electronic structure and macroscopic properties of solids |

17:00-18:00 | Poster Viewing |

**Wednesday 25 June**

Chair: | Chris Eilbeck |

09:30-10:30 | Bob Kohn (New York University) Cloaking by change of variables |

10:30-11:00 | Refreshments |

11:00-12:00 | Gero Friesecke (University of Warwick) A mathematical picture of basic aspects of the periodic table |

12:00-13:00 | Oliver Penrose (Heriot-Watt University) Statisical mechanics of nonlinear elasticity in the system of hard disks |

13:00-14:30 | Lunch |

14:30-19:00 | Afternoon Free |

**Thursday 26 June**

Chair: | Stefan Müller |

09:30-10:30 | François Murat (Université Pierre et Marie Curie (Paris VI))A priori estimate and existence for elliptic problems with subquadratic gradient dependent terms |

10:30-11:00 | Refreshments |

11:00-12:00 | Laszlo Szekelyhidi (Max Planck Institute, Leipzig) The h-principle in fluid mechanics |

12:00-13:00 | Michael Ortiz (California Institute of Technology)Minimum principles for characterizing the trajectories and microstructural evolution of dissipative systems |

13:00-14:30 | Lunch |

Chair: | François Murat |

14:30-15:30 | Bernard Dacorogna (École Polytechnique Fédérale de Lausanne) On the pullback equation |

15:30-16:00 | Refreshments |

16:00-17:00 | Gilles Francfort (Université Paris-Nord) Evolution and approximation in brittle fracture |

17:00-18:00 | Discussions |

Host: | Robin Knops |

19:00-22:30 | Evening Banquet |

**Friday 27 June**

Chair: | Marshall Slemrod |

09:30-10:30 | Robert Pego (Carnegie Mellon University) How to compute the pressure in incompressible viscous flow with no-slip boundary |

10:30-11:00 | Refreshments |

11:00-12:00 | Antonio DeSimone (SISSA, Trieste) Excellent swimmers |

12:00-13:00 | Carsten Carstensen (Humboldt-Universität zu Berlin) Computational challenges in the calculus of variations |

13:00-14:30 | Lunch |

Chair: | Jan Kristensen |

14:30-15:30 | Yann Brenier (Université de Nice-Sophia-Antipolis) Convection, magnetic relaxation and optimal transport |

15:30-16:00 | Refreshments |

16:00-17:00 | Terry Tao (University of California at Los Angeles)Global regularity of wave maps via harmonic map heat flow |

17:00-18:00 | Jack Carr: Close of Meeting |

### Presentations

are very small, their smallness signalled by a small parameter. For such

problems it is customary to discard the small terms, in which case some

or all the variables evolve quasistatically, i.e., through a sequence of

equilibrium states parametrized by time. Formal asymptotic expansions

in the small parameter might exhibit the detailed effects of the parameter.

Rigorous asymptotic justifications, which provide error estimates and

are typically far harder to carry out, are used by those compulsive about

mathematical hygiene. They are valuable, however, when the formal theory

exhibits surprising results.

The purpose of this lecture is to illuminate the material behavior supporting

the asymptotic justification of quasistatic response and to examine

some strange and surprising large-time dynamical behavior when such

justification is not available. These results will be described in the context

of a couple of conceptually simple problems from particle and continuum

mechanics.

The mathematical problems analyzed in this presentation are motivated by

two types of modeling endeavors:

cell motility in fiber matrices and elastic behavior of

body-implantable biomedical devices. Some of these systems are also highly anisotropic and exhibit liquid crystal behavior.

I formulate and analyze boundary value problems arising in the design and implanting of biomedical devices.

I will also discuss time dependent regimes dominated by dissipation. One main difficulty

is the fact that both formulations, Lagrangian and Eulerian are present in the governing systems of partial differential equations. Finally, I will address the early-dynamics regime where chemical potentials dominate over viscosity.

The governing system in such case is hyperbolic with a source term.

Polyconvex minimisation problems allow for the existence of global minimisers with unclear regularity. For arbitrarily small loads in hyperelasticity, global minimisers coincide with local smooth solutions based on the implicit function theorem. In this situation, finite element methods are convergent as well. For more realistic loads, however, even $Gamma$ convergence of finite element approximations is uncertain.

Finite strain elastoplasticity is based on a multiplicative split of elastic and plastic deformations which results in a non-convex mathematical model. In fact, a typical time-step within some incremental description of the elastoplastic evolution results in a non-quasiconvex minimisation problem. Difficulties with the efficient numerical relaxation are discussed for some single-slip elastoplastic benchmark.

Nonlinear minimisation problems allow exciting phenomena such as repelling singularities. The Lavrentiev phenomena is shown to be equivalent to the failure of finite element methods. This motivates non-conforming discrete spaces and/or modified energies. The presentations reviews some solution strategies and a $L^p$ penalisation scheme with some preliminary affirmative results.

The presentation is based on recent joint work with S. Conti, R. Huth, A. Orlando, C. Ortner.

Based on three-dimensional nonlinear elasticity and by using a combination of explicit constructions and general results from differential geometry, we prove that, in agreement with previous heuristic results in the physics literature, the total energy per unit thickness of such folding patterns scales at most as the thickness of the sheet to the power $5/3$. We contrast this scaling with the linear scaling obtained in the case of Dirichlet boundary conditions.

Mathematically we are interested in the critical magnetic field at which the lowest energy state gets into the continuum spectrum, describing a destabilizing effect.

A scaling argument allows us to find a simple variational characterization for the critical magnetic field. We use it in order to compare the general critical magnetic field and the one found when taking into account only states which lie in the lowest Landau level. We do this both analytically and numerically. We find an important discrepancy between the two regimes.

Finally, we will examine other approximations that are popular in the mechanics community and discuss their relevance to fracture evolution.

In the talk I will present a mathematical picture (for the first ten elements H, He, Li, Be, B, C, N, O, F, Ne), based on calculating the many-electron Schroedinger ground state explicitly in a natural scaling limit. In this limit, the nontrivial postulates of Bohr, Hund and Slater (electrons filling orbitals, shell and sub-shell formation,

sub-shell ordering rules such as 2s < 2p, Hund's rules) are seen to emerge in a natural way.

In fact, in a small minority of cases (such as ordering of the excited states of the Carbon atom) our mathematical results differ from the semi-empirical theory (and experiment confirms our predictions). Proofs involve hydrogen atom theory, careful use of the symmetry group SU(2) x SO(3) x Z_2 of the many-electron Schroedinger equation and its representation theory, Fourier analysis, and ideas from quantum chemistry.

Joint work with Ben Goddard (Warwick).

; ; hbox{in} ; {cal D}'(Omega),$$ with $A$ a coercive matrix with bounded coefficients, $alpha_0 geq 0$, $0 leq q leq 2$ and $fin L^m (Omega)$ for some suitable~$m$. This is a model problem, and there are many possible variants of it. In particular, I will consider the case where the right-hand side is only bounded by (but not equal to) $gamma |Du|^q + f(x)$.

In the case where $0 leq q < 1$, existence is classical for $f in H^{-1} (Omega)$. When $gamma$ is large, the case where $q = 1$ and $f in H^{-1} (Omega)$ is difficult but has been solved by G. Bottaro and M.E. Marina in 1973. On the other hand, the case $q = 2$ has been treated by many authors, including in particular in a series of papers by L.~Boccardo, J.-P.~Puel and myself. In a more recent paper, V. Ferone and myself proved the existence of a solution $u$ which further satisfies

$ e^{gamma u} -1 in H^1_0 (Omega)$, and an a priori estimate for such solutions, when %$fin L^{{N} over {2}} (Omega)$.

In this lecture I will mainly report about recent joint work with Nathalie Grenon and Alessio Porretta, the announcement of which has been published in C. R. Acad. Sci. Paris, S'erie I 342 (2006), {23-28}. When $1 + (2 / N) leq q <2$ and $fin L^m(Omega)$ with $m = {{N (q-1)} / {q}}$ (we also solved the case where $1leq q < 1 + {{2}over{N}} $, but I will not discuss it since it uses the notion of renormalized solution), and when either $alpha_0 >0$ or $f$ is sufficiently small in $L^m(Omega)$, we prove the existence of a solution $u$ which enjoys the further regularity $|u|^sigma in H^1_0(Omega)$ with, $sigma = {{(N-2) (q - 1)} / {2(2 - q)}}$. Uniqueness results in this class have also been proved by G. Barles and A. Porretta under certain structure conditions. We moreover prove an a priori estimate for every solution in this class. One of the main interests of our work lies in this a priori estimate, the proof of which is non standard.

incompressible flow with no-slip boundary conditions, that involves the

commutator of the Laplacian and Leray-Helmholtz projection operators.

An estimate for this commutator shows that it is strictly dominated

by the viscous term at leading order. This leads to a number of new developments,

including a well-posedness theorem for an extended Navier-Stokes dynamics

unconstrained by the divergence-free condition, and a number of significant

improvements concerning methods of numerical computation and numerical analysis

for such flows. This is joint work with Jian-Guo Liu (Maryland) and Jie Liu (Irvine).

without an attractive short-range interaction. A `constrained free energy'

is defined, using the standard configurational integral of statistical

mechanics subject to two constraints which relate the set of

configurations we integrate over to a given close-packed reference

configuration of the set of disks under consideration. These constraints

are: (i) if two disks are nearest neighbours in the reference

configuration, then their separation is constrained to be at most sqrt{2}

times the disk diameter and (ii) each disk at the boundary of the

reference configuration is constrained to stay within a specified distance

from a specified ``anchor point". It is shown that, if the anchor points

of the boundary particles are related to their reference positions by a

constant deformation matrix A belonging to a certain open set (whose

closure comprises all 2 x 2 matrices which, when applied to the reference

configuration, produce a configuration that is compatible with the

constraint (i) and also with the condition that disks must not overlap),

then the free energy per particle, related to the partition function in

the usual way, has a well-defined thermodynamic limit. It is also shown

that the free energy per particle is a Lipschitz-continuous function of

the matrix A and that it satisfies the integral inequality characterizing

quasi-convex functions.

We will discuss what is known about this problem and its connections to other questions. The problem is open in the general 3d situation, but results can be obtained in some interesting special cases.

### Participants

Name | Institution |
---|---|

Stefan, Adams | University of Warwick |

Stuart, Antman | University of Maryland |

John, Ball | Heriot-Watt University |

Peter, Bates | Michigan State University |

Margaret, Beck | Heriot-Watt University |

Kaushik, Bhattacharya | California Institute of Technology |

Natalia, Bochkina | University of Edinburgh |

David, Bourne | University of Glasgow |

Yann, Brenier | Ecole Polytechnique |

Ken, Brown | Heriot-Watt University |

Martin, Burns | University of Strathclyde |

Carme, Calderer | University of Minnesota |

Yves, Capdeboscq | University of Oxford |

Anthony, Carbery | University of Edinburgh |

Jack, Carr | Heriot-Watt University |

Carsten, Carstensen | Humboldt-Universität zu Berlin |

Isaac, Chenchiah | University of Bristol |

Kirill, Cherednichenko | University of Bath |

Miroslav, Chlebik | University of Sussex |

Adam, Chmaj | Warsaw University of Technology |

Philippe G., Ciarlet | City University of Hong Kong |

Sergio, Conti | Universität Bonn |

Daniel, Coutand | Heriot-Watt University |

Elaine, Crooks | University of Swansea |

Bernard, Dacorogna | École Polytechnique Fédérale de Lausanne |

Constantine, Dafermos | Brown University |

Fordyce, Davidson | University of Dundee |

Penny, Davies | University of Strathclyde |

Antonio, DeSimone | Scuola Internazionale Superiore di Studi Avanzati |

Wlodzimierz, Domanski | Polish Academy of Sciences |

Patrick, Dondl | Durham University |

Robert, Douglas | Aberystwyth University |

Dugald, Duncan | Heriot-Watt University |

Angel, Duran | University of Valladolid |

J. Chris, Eilbeck | Heriot-Watt University |

Maria J, Esteban | Université Paris-Dauphine |

Jozsef, Farkas | University of Stirling |

Gilles A., Francfort | Université Paris-Nord |

Gero, Friesecke | TU Munich |

Michael, Grinfeld | University of Strathclyde |

Mark, Haskins | Imperial College London |

Duvan, Henao | University of Oxford |

Viet Ha, Hoang | Nanyang Technological University |

Gunnar, Hornig | University of Dundee |

Peter, Hornung | University of Bath |

Cheng-Hsiung, Hsu | National Central University,Taiwan |

Wen-Guei, Hu | National Chiao Tung University |

Richard, James | University of Minnesota |

Abdelouahab, Kadem | University of Setif |

Agnieszka, Kalamajska | University of Warsaw |

Biswajit, Karmakar | Max Planck Institute |

Abdul Rahim, Khan | King Fahd University of Petroleum and Minerals |

Robin, Knops | Heriot-Watt University |

Gabriel, Koch | University of Chicago |

Robert V., Kohn | New York University |

Gerasim, Kokarev | University of Leeds |

Michalis, Kontovourkis | Max Planck Institute |

Konstantinos, Koumatos | University of Oxford |

Jan, Kristensen | University of Oxford |

Andrew, Lacey | Heriot-Watt University |

Ben, Leimkuhler | University of Edinburgh |

Gabriel, Lord | Heriot-Watt University |

Alex, Mahalov | Arizona State University |

Emmanuel, Maitre | Université de Grenoble |

Maroje, Marohnic | University of Zagreb |

Karsten, Matthies | University of Bath |

Christof, Melcher | University of Oxford |

Roger, Moser | University of Bath |

Benson, Muite | University of Oxford |

Stefan, Müller | Max Planck Institute |

François, Murat | Université Pierre et Marie Curie (Paris VI) |

Camil, Muscalu | Cornell University |

Sergiy, Nesenenko | Darmstadt University of Technology |

Stefan, Neukamm | TU München |

Igor, Neygebauer | National University of Rwanda |

Barbara, Niethammer | University of Bonn |

Michael, Ortiz | California Institute of Technology |

Christoph, Ortner | University of Warwick |

Felix, Otto | Max Planck Institute for Mathematics in the Sciences, Leipzig |

Iryna, Pankratova | Narvik University College |

Robert, Pego | Carnegie Mellon University |

Oliver, Penrose | Heriot-Watt University |

Mircea, Petrache | Scuola Normale Superiore |

Andrey, Piatnitski | Narvik University College |

Andrey, Rekalo | Heriot-Watt University |

David, Rule | University of Edinburgh |

Bryan, Rynne | Heriot-Watt University |

Witold, Sadowski | University of Warwick |

Dominique (Nick), Schryvers | University of Antwerp |

Yasemin, Sengul | University of Oxford |

Gregory, Seregin | University of Oxford |

Hélia, Serrano | Universidad de Castilla-La Mancha |

Eugene, Shargorodsky | King's College London |

Michael, Shearer | North Carolina State University |

Michael, Singer | University College London |

Jeyabal, Sivaloganathan | University of Bath |

Valeriy, Slastikov | University of Bristol |

Marshall, Slemrod | University of Wisconsin |

Wolfgang, Staubach | Heriot-Watt University |

Philipp Emanuel, Stelzig | TU München/Università di Trento |

Pawel, Strzelecki | University of Warsaw |

Endre, Suli | University of Oxford |

Vladimir, Sverak | University of Minnesota |

László, Székelyhidi | Leipzig University |

Ali, Taheri | University of Sussex |

Josip, Tambaca | University of Zagreb |

Antoine, Tambue | Heriot-Watt University |

Terence, Tao | University of California at Los Angeles |

John, Toland | Isaac Newton Institute for Mathematical Sciences |

Basang, Tsering | University of Oxford |

Igor, Velcic | University of Zagreb |

David, Wagner | University of Houston |

Stephen, Watson | University of Glasgow |

Tzi-Sheng, Yang | Tunghai University |

Ting-Hui, Yang | Tamkang University |

Arghir, Zarnescu | University of Sussex |

Kewei, Zhang | University of Swansea |