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Seminars 2007-2008

Models for dynamic fracture based on Griffith's criteria

Initial boundary value problem for 2d viscous Boussinesq equations

We prove the global regularity for 2d viscous Boussinesq Equations in a bounded domain with no-slip boundary condition. The uniform estimate on kinetic energy is obtained as a by product.

The composite membrane problem

We study the problem of building a body of given mass and shape out of materials of varying density, so as to minimize the first Dirichlet eigenvalue of the body. We study the regularity of the eigenfunction and the regularity of the free boundary

Formulation of an Unconditionally Stable Space--Time Discontinuous Galerkin Method For Linear Thermo-Elasto-Dynamics With Propagating Weak Discontinuities

In this talk a DG space-time FEM is presented that can correctly represent moving and stationary singular surfaces in boundary value problems of fully coupled thermo-elasto-dynamics. The proposed formulation is shown to be consistent and unconditionally stable. The key aspect of the formulation is that numerical stability is achieved by a physically-based approach consisting in enforcing the jump conditions of the balance of linear momentum and the balance of energy. The method's convergence rates are demonstrated using some exact solutions of two- and three-dimensional space-time boundary value problems. Solutions to boundary value problems with moving discontinuities are presented in which the degree of thermo-mechanical coupling is varied to show how the system's response in terms of stress, temperature, and energy release rate is affected.

Navier-Stokes with no-slip boundary—Analysis and computation based on a commutator formula for pressure

Surfaces of prescribed p-mean curvature in the Heisenberg group

The role of Q-curvature in conformal geometry

Quadruple junction solution in R^3

In this talk, I will discuss the quadruple junction solutions in the entire three dimensional space to a vector-valued Allen-Cahn equation which models multiple phase separation.

The solution is the basic profile of the local structure near a quadruple junction in three dimensional crystalline material under the generalized Allen-Cahn model, and is the three dimensional counterpart of triple junction solution which is two dimensional.

I will start with one dimensional heteroclinic solutions, and describe how we can construct higher dimensional solutions from the lower dimensional ones, and explain the complications and difficulties in constructing such solution in three dimensions.

On some concentration phenomena for a nonlinear autonomous elliptic system

Existence of Capillary minimal disks inside small tubes in Riemannian manifolds

A contact structure which describes the functional geometry of the visual cortex

In collaboration with A. Sarti and J. Petitot, we propose a model of the functional architecture of the primary visual cortex V1 as a principal fiber bundle. The 2-dimensional retinal plane is the base manifold and the secondary variables (which can be orientation or scale) constitute the vertical fibers over each point. In the description of the odd simple cells we consider only the orientation, and the resulting structure is a contact structure. When we consider the even cells we add the scale variable, and the total space is equipped with a symplectic structure, which naturally extends the previous one. Both geometrical structure are implemented by the long range horizontal connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures such as the medial axis of a shape.

Semiclassical states for the nonlinear Schrodinger-Poisson system: concentration on spheres

On concentration of positive bound states for the Schrodinger-Poisson problem with an external potential and a density charge

Viscoleastic relaxation for wrinkled membranes

Theory and Simulation of a Brittle Damage Model in Thermoelastodynamics

Finite Element implementation of Cam-Clay models

A Zermelo Navigation Problem with Applications to Lindblad Equations

Hardy-Sobolev-Maz'ya inequalities and hyperbolic symmetry

(tba)

Permutations.

ICTP/SISSA Joint Colloquium in Mathematics

Random permutations of N elements have peculiar statistics of the lengths of the cycles (discovered by V.L. Gontcharov, 1942).

The number of the cycles of a random permutation of N points grows with N (in the mean) as 1+ 1/2 +1/3 + … + 1/N ~ 0,58 + ln N.

The Fibonacci cat map is the action of the matrix [] permuting the n^2 points of the finite torus Z/nZ x Z/nZ. Such algebraically defined permutations have quite different statistics (from that of the random ones).

Example: Some permutations of 100 points have periods exceeding 230 million.

The length of the period of the cat map, permuting 150x150 pixels, is only 300. The study of similar examples (begun in the 1990s by I. Persival and F. Dyson) has led to unexpected mathematical theorems.

Example: The number of those permutations of 2N points, all whose cycles are of even length, is a square of an integer, namely, of (2N-1)!! =1.3.5…..(2N-1).

Extremals for Hardy­Sobolev type inequalities: the influence of the curvature.

(Joint work with N.Ghoussoub)

We consider the optimal Hardy­Sobolev inequality on a smooth bounded domain of the Euclidean space. Roughly speaking, this inequality lies between the Hardy inequality and the Sobolev inequality. We address the questions of the value of the optimal constant and the existence of non­trivial extremals attached to this inequality. When the singularity of the Hardy part is located on the boundary of the domain, the geometry of the domain plays a crucial role: in particular, the convexity and the mean curvature are involved in these questions. The main difficulty to encounter is the possible bubbling phenomenon. We describe precisely this bubbling through refined concentration estimates. An offshot of these techniques allow us to provide general compactness properties for nonlinear equations, still under curvature conditions for the boundary of the domain.

Rolling balls and Octonions.

Arnold Seminar at ICTP, Trieste

We consider mysterious hidden symmetries of a classical non-holonomic kinematic model. The problem was not stated by Vladimir Arnold but, I hope, its study is in the spirit of the Arnold school. Moreover, I hope that some participants of the seminar would help to find a better geometric interpretation of these symmetries.

Straight lines of a plane and real algebraic geometry.

ICTP/SISSA Joint Colloquium in Mathematics

Which values may attain the number $M$ of the connected components of the complement to $n$ distinct straight lines in the real plane?

For $n=4$ the possible numbers of components are (5, 8, 9, 10, 11), but the general problem for $n$ straight lines is unsolved.

Namely, between the minimal number of components, $n+1$, and the maximal one, $(n^2+n+2)/2$, there are gaps formed by the unattainable numbers.

The first gap is $n+1<M<2n$: these values are unattainable. For the $k$-th gap the stable boundaries $a_k(n)<M< b_k(n)$ are known, providing the answer for the stable case (where the number of straight lines is sufficiently high, $n>C(k)$). For the unstable values of $n$, the $k$-th gap may be smaller than the stable answer. Its exact boundary is unknown even for the 3-rd gap (where the stability starts from $n=14$ straight lines).

In real algebraic geometry even the simplest problems are difficult (and even the contribution of Hilbert to his 16-th problem, discussing these questions, was wrong).

Existence of hyper-surfaces with large Constant Mean Curvature and Free Boundaries.

Let $\Omega$ an open bounded smooth domain of $\R^{n+1}$. We establish existence of family of stationary sets for the relative Perimeter functional of $\Omega$ with small volume constrained which "condense" to a sub-manifold $K^k$ of the boundary of $\Omega$ with dimension $k<n$. We will discuss the cases: k=0 and k>0. The Latter case is a joint work with Mahmoudi F.

Flow and Stability of Liquids at Interfaces: Liquid Slip and Pattern Formation.

In recent years there has been an explosion of interest in the behaviour of liquids in confined geometries, especially due to the exciting developments in the fields of microfluidic devices and of nanotechnology. In this talk I will review results of my research performed on liquid/solid interfaces, from studies of the boundary conditions of flow of liquids at solid surfaces, to pattern formation in thin polymer films. Finally, I will introduce a new strategy to produce surfaces with spatially localised features of controlled surface chemistry, with potential applications to the patterning of proteins and cells in biology and biotechnology.

Constant mean curvature foliations of asymptotically hyperbolic spaces.

Let (M,g) be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on the boundary of M and constant mean curvature foliations in some neighbourhood of infinity in M. There is a subtle interplay between the precise terms in the expansion for the metric g and various properties of the foliation. This is a joint work with R. Mazzeo.

Curvature flows on four manifolds with boundary.

Last modified: 15 January 2009

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