Past Applied Analysis and Mechanics Seminar

Two dimensional minimal cones were fully classified by Jean Taylor in the mid

70's.  In joint work with G. David and T. De Pauw we prove that a closed

set which is close to a minimal cone at all scales and at all locations is

locally a bi-Hoelder image of a minimal cone.  This result is analogous to

Reifenberg's disk theorem.  A couple of applications will be discussed.

In this talk we will discuss a theory of divergence-measure fields and related

geometric measures, developed recently, and its applications to some fundamental

issues in mathematical continuum physics and nonlinear conservation laws whose

solutions have very weak regularity, including hyperbolic conservation laws,

degenerate parabolic equations, degenerate elliptic equations, among others.

For atomistic (and related) material models, global minimization

gives the wrong qualitative behaviour; a theory of equilibrium

solutions needs to be defined in different terms. In this talk, a

process based on gradient flow evolutions is presented, to describe

local minimization for simple atomistic models based on the Lennard-

Jones potential. As an application, it is shown that an atomistic

gradient flow evolution converges to a gradient flow of a continuum

energy, as the spacing between the atoms tends to zero. In addition,

the convergence of the resulting equilibria is investigated, in the

case of both elastic deformation and fracture.

A coupled cell system is a collection of interacting dynamical systems.
Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: How much of the qualitative dynamics observed in coupled cells is the product of network architecture and how much depends on the specific equations?

The ideas will be illustrated through a series of examples and theorems. One theorem classifies spatio-temporal symmetries of periodic solutions and a second gives necessary and sufficient conditions for synchrony in terms of network architecture.

/notices/events/abstracts/applied-analysis/tt05/Heywood.pdf

We derive a two-dimensional compressible elasticity model for thin elastic sheets as a Gamma-limit of a fully three-dimensional incompressible theory. The energy density of the reduced problem is obtained in two

steps: first one optimizes locally over out-of-plane deformations, then one passes to the quasiconvex envelope of the resulting energy density. This work extends the results by LeDret and Raoult on smooth and finite-valued energies to the case incompressible materials. The main difficulty in this extension is the construction of a recovery sequence which satisfies the nonlinear constraint of incompressibility pointwise everywhere.

This is joint work with Sergio Conti.

Bacteriophage T4 is a virus that attacks bacteria by a unique mechanism. It

lands on the surface of the bacterium and attaches its baseplate to the cell

wall. Aided by Brownian motion and chemical bonding, its tail fibres stick to

the cell wall, producing a large moment on the baseplate. This triggers an

amazing phase transformation in the tail sheath, of martensitic type, that

causes it to shorten and fatten. The transformation strain is about 50%. With a

thrusting and twisting motion, this transformation drives the stiff inner tail

core through the cell wall of the bacterium. The DNA of the virus then enters

the cell through the hollow tail core, leading to the invasion of the host.

This is a natural machine. As we ponder the possibility of making man-made

machines that can have intimate interactions with natural ones, on the scale of

biochemical processes, it is an interesting prototype. We present a mathematical

theory of the martensitic transformation that occurs in T4 tail sheath.

Following a suggestion of Pauling, we propose a theory of an active protein

sheet with certain local interactions between molecules. The free energy is

found to have a double-well structure. Using the explicit geometry of T4 tail

sheath we introduce constraints to simplify the theory. Configurations

corresponding to the two phases are found and an approximate formula for the

force generated by contraction is given. The predicted behaviour of the sheet is

completely unlike macroscopic sheets. To understand the position of this

bioactuator relative to nonbiological actuators, the forces and energies are

compared with those generated by inorganic actuators, including nonbiological

martensitic transformations. Joint work with Wayne Falk, @email

Wayne Falk and R. D. James, An elasticity theory for self-assembled protein

lattices with application to the martensitic transformation in Bacteriophage T4

tail sheath, preprint.

K. Bhattacharya and R. D. James, The material is the machine, Science 307

(2005), pp. 53-54.

We use the partial differential inclusion method to establish existence of

infinitely many weak solutions to the one-dimensional version of the

Perona-Malek anisotropic diffusion model in the theory of image processing. We

consider the homogeneous Neumann problem as the model requires.

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An important class of nonlinear parabolic equations is the class of quasi-linear equations, i.e., equations with a leading second-order (in space) linear part (e.g., the Laplacian) and a nonlinear part which depends on the first-order spatial derivatives of the unknown function. This class contains the Navier-Stokes system of fluid dynamics, as well as "viscous" versions (or "regularized") of the Hamilton-Jacobi equation, nonlinear hyperbolic conservation laws and more. The talk will present various recent results concerning existence/uniqueness (and nonexistence/nonuniqueness) of global solutions. In addition, a new class of "Bernstein-type" estimates of derivatives will be presented. These estimates are independent of the viscosity parameter and thus lead to results concerning the "zero-viscosity" limit.

/notices/events/abstracts/applied-analysis/ht05/gudrun.pdf

/notices/events/abstracts/applied-analysis/ht05/pohozaev.shtml

/notices/events/abstracts/applied-analysis/ht05/ogden.shtml

Coherent structures, or defects, are interfaces between wave trains with

possibly different wavenumbers: they are time-periodic in an appropriate

coordinate frame and connect two, possibly different, spatially-periodic

travelling waves. We propose a classification of defects into four

different classes which have all been observed experimentally. The

characteristic distinguishing these classes is the sign of the group

velocities of the wave trains to either side of the defect, measured

relative to the speed of the defect. Using a spatial-dynamics description

in which defects correspond to homoclinic and heteroclinic orbits, we then

relate robustness properties of defects to their spectral stability

properties. If time permits, we will also discuss how defects interact with

each other.

Marstrand's Theorem is a one of the classic results of Geometric Measure Theory, amongst other things it says that fractal measures do not have density. All methods of proof have used symmetry properties of Euclidean space in an essential way. We will present an elementary history of the subject and state a version of Marstrand's theorem which holds for spaces whose unit ball is a polytope.

While the classification of crystals made up by just one atom per cell is well-known and understood (Bravais lattices), that for more complex structures is not. We present a geometric way classifying these crystals and an arithmetic one, the latter introduced in solid mechanics only recently. The two approaches are then compared. Our main result states that they are actually equivalent; this way a geometric interpretation of the arithmetic criterion in given. These results are useful for the kinematic description of solid-solid phase transitions. Finally we will reformulate the arithmetic point of view in terms of group cohomology, giving an intrinsic view and showing interesting features.