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Seminars 2008-2009

Linear Gamma-limits of multiwell energies in nonlinear elasticity

The Poincare-Einstein programme and overdetermined PDE

A compact manifold with boundary is said to have a Poincare-Einstein structure if its interior is equipped with a negative curvature Einstein metric, in terms of which the boundary is suitably ``at infinity''. A central problem is to relate the conformal geometry of this boundary to the Riemannian structure of the interior, and this is linked to the ideas behind Maldacena's AdS/CFT correspondence in String theory. There is a natural approach to aspects of this problem via conformal geometry, a certain overdetermined PDE and its prolongations. This approach also leads to a natural way to extend the programme, and new problems in geometric analysis.

Systems analysis of biological networks under uncertainty

A Sup + Inf inequality for Liouville type equations with weights

Inequalities for Vector Fields

We prove a Bourgain Brezis inequality for Vector Fields, and give volume comparison theorems and isoperimetric inequalities on CR manifolds.

Variational methods for the Willmore functional

The Willmore functional of a two-dimensional surface is the $L^2$ integral of the mean curvature. After briefly presenting some basic geometric features, we will discuss the gradient flow of the functional including a convergence result from [3]. In the second part we present a compactness theorem for sequences of surfaces with specific bounds on the Willmore energy. This result was used recently to construct minimizers of the Willmore functional with prescribed conformal type.

1. Kuwert, E., Sch\"atzle, R., The Willmore flow with small initial energy, J. Differential Geom. 57 (2001), 409--441.
2. Kuwert, E., Sch\"atzle, R., Gradient flow for the Willmore functional, Comm. Anal. Geom. 10 (2002), 307--339.
3. Kuwert, E., Sch\"atzle, R., Removability of point singularities of Willmore surfaces, Ann. of Math. 160 (2004), 315--357.
4. Kuwert, E., Sch\"atzle, R., Closed surfaces with bounds on their Willmore energy, Preprint 2008, CRM Ennio De Giorgi, Pisa.
5. Kuwert, E., Sch\"atzle, R., Minimizers of the Willmore functional with prescribed conformal type, Preprint 2008.

Nonlocal character of the reduced theory of thin films with higher order perturbations

Levi curvatures and related problems (ICTP/SISSA Joint Colloquium in Mathematics)

The Levi curvatures of a real hypersurfaces of C^{n+1} are related to the Levi form the way the usual Mean or Gauss curvatures are related to the real Hessian form. They were implicitly introduced by Bedford and Gaveau in 1978, and more explicitly, in the case n=1, by Tomassini in 1988. Later on, Montanari and Lanconelli extended these notions to any dimension, following the lines of the works by Caffarelli, Nirenberg and Spruck on the real Monge-Ampere equations. Montanari and Lanconelli showed that, for hypersurface graphs on real functions, the Levi curvatures can be expressed in terms of second order fully nonlinear PDO's with an underlying sub-Riemannian structure. The properties of these structures allowed one to obtain:

1. A strong comparison principle for hypersurfaces having the same Levi curvatures;
2. Isoperimetric inequalities;
3. Alexandrov-type sphere theorems;
4. Existence theorems of Lipschitz-continuous viscosity solutions to the prescribed Levi-curvature problems;
5. (only for n=1) Smoothness of the Lipschitz-continuous viscosity solutions.

The results in 3. and 5. only give partial answers to problems which are still largely open.

Classical and new variational models for dislocations

On existence and behavior of radial minimizers for the Schrodinger-Poisson-Slater problem

Soluzioni deboli per il flusso di curvatura media: barriere minime

Linear time optimality

As it is well known, the time optimal control problem for a linear system with one-dimensional control is reduced to the certain algebraic system w.r.t. the optimal time and the switches of the optimal (bang-bang) control. In the particular “integrator” case $\dot x_1 = u, \dot x_k = x_{k−1}, k=2,...,n$, this system (of n equations w.r.t. n variables) is polynomial of a very special form. Such systems were studied much earlier within the moment problem theory. In particular, the time optimal problem for the integrator system is reformulated as the (in fact, nonclassical) power Markov moment problem on the minimal possible interval.

The exact analytic solution for this problem and some generalizations will be presented. The lecture is mainly based on the results by V.I.Korobov and G.M.Sklyar (Kharkov National University) obtained since 1987.

Relaxation results for nematic elastomers

First Order Implicit ODE's and Their Applications

Conservation laws in conformal geometric analysis (ICTP/SISSA Joint Colloquium in Mathematics)

In this talk we aim to present regularity and compactness properties for critical points to conformally invariant lagrangian playing a special role in differential geometry : harmonic maps into riemannian and pseudo-riemannian manifolds, minimal surfaces, prescribed mean curvature surfaces, Willmore surfaces, 1/2-harmonic maps and bi-harmonic maps into manifolds, etc.

We will isolate common features to all these objects and in particular show the existence of "hidden" conservation laws which do not enter in the classical framework of Noether's Theorem.

We will recall some fundamental results of the theory of integrability by compensation and we will show how, combined with the existence of conservation laws, this theory permits to overcome a number of analysis difficulties of these particular elliptic partial differential systems.

Some Remarks on Regularity for Embedded Curves Moving by Mean Curvature in the Plane

We will discuss some ideas leading to an alternative line of proof of the Gage-Hamilton-Grayson theorem that an embedded closed curve in the plane, moving by mean curvature, does not develop singularities during the flow and shrinks to a point in finite time.

Some of these techniques also extend to the analysis of the flow of surfaces in $R^3$.

Flow regulation for synthetic genetic circuits

Flow control is a fundamental feature for the correct performance of large scale networks, of which familiar examples are the Internet, power grids or even pipe networks. In the biological world, complex cellular pathways rely as heavily on a regulated flow of nucleic acids, transcription factors and other metabolites. It is therefore important to explore and understand all the possibilities for production and degradation rate control in biochemical components. Two designs to regulate and match the RNA production of two in vitro synthetic genetic circuits (J. Kim 2006) will be considered. In particular, these architectures are based on self inhibition and cross activation mechanisms that can dynamically change the fraction of actively transcribing DNA strands and correctly respond to changes in the chemical environment.

The two regulatory architectures are numerically analyzed trough a first order model derived from the set of occurring reactions. Most importantly, experimental results show the effectiveness of the proposed feedback loops in regulating the RNA out-flow of the circuits.

Colloquium: Geometric Flows

We introduce a class of evolution equations which depend on the geometric properties of the objects involved. Such motions, as for example the mean curvature flow, the inverse mean curvature flow and the Ricci flow have important roles in both applied and pure subjects, like physical phenomena with surface tension, general relativity and differential geometry. We will review some of the main motivations/applications for the study, and describe some of the mathematical features of the equations.

The seminar will be preceded by a get-together at 4.10 p.m. in front of room D for tea and coffee.

Spectral radius, index estimates for Schrodinger operators and geometric applications

Non-linear aspects of Calderon-Zygmund theory

Classical Calderon-Zygmund theory allows to retrive in a sharp way the integrability properties of solutions to linear elliptic and parabolic equations starting from that of the given data. I will outline some non-linear analogs of this valid for quasilinear operators, for example those of p-Laplacean type.

Cut locus in sub-Riemannian problem on the group of motions of a plane

Consider a mobile robot in the plane that can move forward and backward, and rotate around itself. The state of the robot is described by coordinates (x,y) of its center of mass and by angle of orientation theta. Given an initial and a terminal state of the robot, one should find the shortest path from the initial state to the terminal one, when the length of the path is measured in the space (x,y,theta).

Such a problem is formalized as a left-invariant sub-Riemannian problem on the group of motions of a plane. The talk will be devoted to recent results on this optimal control problem obtained by geometric techniques:

• sub-Riemannian geodesics are parametrized by Jacobi's functions,
• the Maxwell points corresponding to discrete symmetries generated by reflections of pendulum are described,
• the first conjugate time is bounded by Maxwell times,
• the cut time is proved to be equal to the first Maxwell time,
• the global structure of the exponential mapping and optimal synthesis are described,
• the cut locus is parametrized.

Variational Analysis via discrete approximations in optimal control

Modern variational analysis can be viewed as a certain outgrowth of the classical calculus of variations, optimal control and mathematical programming, where perturbation/approximation methods and stability/sensitivity issues play a crucial role.

In this talk we discuss recent advances in variational analysis and its applications in dynamic optimization models governed by nonconvex differential inclusions in both finite and infinite dimensions, which happen to be a natural framework for the unifying study of various problems in the calculus of variations and optimal control.

We discuss an approach to such problems based on finite-difference/discrete approximations of differential systems and thus related to both numerical and theoretical issues in optimization and control. Developing this approach, we reduce continuous-time optimal control problems to sequences of constrained mathematical programs of a special type, which are comprehensively studied by using modern methods of variational analysis and generalized differentiation. The main results justify well-posedness/stability of discrete approximations and establish necessary optimality conditions of Euler-Lagrange and Weierstrass-Pontryagin types for nonconvex differential inclusions employing advanced tools of variational analysis.

Analysis of variational models for nematic liquid crystal elastomers

The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells

A longstanding problem in the mathematical theory of elasticity is to predict theories of lower-dimensional objects (such as rods, plates or shells), subject to mechanical deformations, starting from the 3d nonlinear theory. For plates, a recent effort has lead to rigorous justification of a hierarchy of such theories (membrane, Kirchhoff, von Karman). For shells, despite extensive use of their ad-hoc generalizations present in the engineering applications, much less is known from the mathematical point of view.

In this talk, I will discuss the limiting behavior (using the notion of Gamma-limit) of the 3d nonlinear elasticity for thin elliptic shells, as their thickness h converges to 0, under the assumption that the elastic energy of deformations scales like h^\beta, with 2<\beta<4.

Two major ingredients of the proofs are: the density of smooth isometries of Sobolev first order isometries, and a result on matching smooth infinitesimal isometries with exact isometric immersions on smooth elliptic surfaces.

Coherent control of quantum information devices

The development of new technologies at scales approaching the quantum regime is driving new theoretical and experimental research on control in quantum systems. The implementation of quantum control would have an enormous impact on a wide range of fields, in particular in quantum information processing and precision metrology.

In this talk I will present the application of coherent control techniques to the electronic and nuclear spins associated with Nitrogen-Vacancy (NV) centers in diamond. These systems have emerged as excellent candidates for quantum information processing, since they can be optically polarized and detected, and present good coherence properties even at room temperature. I will first show how this solid state system can be used as the building block of a scalable architecture for quantum computation or communication. Then I will present a novel approach to magnetometry, based on NV centers, that takes advantage of coherent control techniques and the confinement of the sensing spins into a sample of nanometer dimensions. The resulting magnetic sensor is projected to yield an unprecedented combination of ultra-high sensitivity and spatial resolution, with the potential of exciting applications in bio-science, materials science, and single electronic and nuclear spin detection.

Quasistatic crack growth in finite elasticity with non-interpenetration

Semicontinuity in SBV: a recent result by Fusco-Leone-March-Verde

A survey on Weak KAM Theorem

Last modified: 12 November 2010

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