Past Applied Analysis and Mechanics Seminar

On parabolic and elliptic equations with VMO coefficients.

Abstract: An L_p-theory of divergence and non-divergence form elliptic and parabolic equations is presented.

The main coefficients are supposed to belong to the class VMO_x, which, in particular, contains all functions independent of x.

Weak uniqueness of the martingale problem associated with such equations is obtained

The discrete structure of the ground states of a spin system is often neglected by averaging on a mesoscopic scale and thus capturing the main features of the model while simplifying its analysis. In many cases this procedure is not rigorous and not even well understood. In this talk we show that the coarse graining procedure for the XY (N-dimensional, possibly anysotropic) spin type model can be made rigorous by using Gamma-convergence. In the two-dimesional case we show how it is possible to address the same problem for a model with long-range interactions. Finally we discuss several possible developments and present some open problems.

Electrical impedance tomography (EIT) technique has been an active research topic since the early 1980s. In EIT, one measures the boundary voltages due to multiple injection currents to reconstruct images of the conductivity distribution. However, these boundary voltages are insensitive to a local change of the conductivity distribution and the relation between them is highly nonlinear. Medical imaging has been one of the important application areas of EIT. Indeed, biological tissues have different electrical properties that change with cell concentration, cellular structure, and molecular composition. Such changes of electrical properties are the manifestations of structural, functional, metabolic, and pathological conditions of tissues, and thus provide valuable diagnostic information. Since all the present EIT technologies are only practically applicable in feature extraction of anomalies, improving EIT calls for innovative measurement techniques that incorporate structural information. The core idea of the approach presented in this talk is to extract more information about the conductivity from data that has been enriched by coupling the electric measurements to localized elastic perturbations. More precisely, we propose to perturb the medium during the electric measurements, by focusing ultrasonic waves on regions of small diameter inside the body. Using a simple model for the mechanical effects of the ultrasound waves, we show that the difference between the measurements in the unperturbed and perturbed configurations is asymptotically equal to the pointwise value of the energy density at the center of the perturbed zone. In practice, the ultrasounds impact a spherical or ellipsoidal zone, of a few millimeters in diameter. The perturbation should thus be sensitive to conductivity variations at the millimeter scale, which is the precision required for breast cancer diagnostic. The material presented in this talk concerning the imaging by perturbation approach, is based on a joint work with Habib Ammari, Eric Bonnetier, Michael Tanter & Mathias Fink and on an ongoing collaboration with Frédéric de Gournay, Otared Kavian and Jérôme Fehrenbach. I will also discuss recent results concerning perturbation of asymptotically small volume fraction which are based on joint works with Michael Vogelius.

  In this talk we will discuss some recent developments in the study of nonlinear partial differential equations of mixed type, including the mixed parabolic-hyperbolic type and mixed elliptic-hyperbolic type. Examples include nonlinear degenerate diffusion-convection equations and transonic flow equations in fluid mechanics, as well as nonlinear equations of mixed type in a fluid mechanical formulation for isometric embedding problems in differential geometry. Further ideas, trends, and open problems in this direction will be also addressed.  

  The intermediate state of a type-I superconductor is a classical example of energy-driven pattern-formation, first studied by Landau in 1937. Mathematically this can be modeled by a nonconvex functional with a singular perturbation, which physically represents the surface energy. In this talk I shall discuss how a combination of interpolation inequalities and explicit constructions permits to determine the scaling of the minimal energy with respect to the relevant material parameters, and therefore to predict a phase diagram for the observed microstructure. This talk is mainly based on joint work with Rustum Choksi, Robert V. Kohn, and Felix Otto.    

 

For a given Lipschitz graph over a subspace without rank-one matrices with

reasonably small Lipschitz constant, we construct quasiconvex functions of

quadratic growth whose zero sets are exactly the Lipschitz graph by using a

translation method. The gradient of the quasiconvex function is strictly

quasi-monotone. When the graph is a smooth compact manifold, the quasiconvex

function equals the squared distance function near the graph.

The corresponding variational integrals satisfy the Palais-Smale compactness

condition under the homogeneous natural boundary condition.

 

/notices/events/abstracts/applied-analysis/ht07/otto.pdf