Gibson 1st Floor SR

We study minimizers of the Ginzburg-Landau (GL) functional \[E_\epsilon(u):=\frac{1}{2}\int_A |\nabla u|^2 + \frac{1}{4\epsilon^2} \int_A(1-|u|^2)^2\] for a complex-valued order parameter $u$ (with no magnetic field). This functional is of fundamental importance in the theory of superconductivity and superuidity; the development of these theories led to three Nobel prizes. For a $2D$ domain $A$ with holes we consider “semistiff” boundary conditions: a Dirichlet condition for the modulus $|u|$, and a homogeneous Neumann condition for the phase $\phi = \mathrm{arg}(u)$. The principal

result of this work (with V. Rybalko) is a proof of the existence of stable local minimizers with vortices (global minimizers do not exist). These vortices are novel in that they approach the boundary and have bounded energy as $\epsilon\to0$.

In contrast, in the well-studied Dirichlet (“stiff”) problem for the GL PDE, the vortices remain distant from the boundary and their energy blows up as

$\epsilon\to 0$. Also, there are no stable minimizers to the homogeneous Neumann (“soft”) problem with vortices.

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Next, we discuss more recent results (with V. Rybalko and O. Misiats) on global minimizers of the full GL functional (with magnetic field) subject to semi-stiff boundary conditions. Here, we show the existence of global minimizers with vortices for both simply and doubly connected domains and describe the location of their vortices.

In the operation of high frequency resonators in micro electromechanical systems (MEMS)there is a strong need to be able to accurately determine the energy loss rates or alternativelythe quality of the resonance. The resonance quality is directly related to a designer’s abilityto assemble high fidelity system response for signal filtering, for example. This hasimplications on robustness and quality of electronic communication and also stronglyinfluences overall rates of power consumption in such devices – i.e. battery life. Pastdesign work was highly focused on the design of single resonators; this arena of work hasnow given way to active efforts at the design and construction of arrays of coupledresonators. The behavior of such systems in the laboratory shows un-necessarily largespread in operational characteristics, which are thought to be the result of manufacturingvariations. However, such statements are difficult to prove due to a lack of availablemethods for predicting resonator damping – even the single resonator problem is difficult.The physical problem requires the modeling of the behavior of a resonant structure (or setof structures) supported by an elastic half-space. The half-space (chip) serves as a physicalsupport for the structure but also as a path for energy loss. Other loss mechanisms can ofcourse be important but in the regime of interest for us, loss of energy through theanchoring support of the structure to the chip is the dominant effect.

The construction of a basic discretized model of such a system leads to a system ofequations with complex-symmetric (not Hermitian) structure. The complex-symmetryarises from the introduction of a radiation boundary conditions to handle the semi-infinitecharacter of the half-space region. Requirements of physical accuracy dictate rather finediscretization and, thusly, large systems of equations. The core to the extraction of relevantphysical performance parameters is dependent upon the underlying modeling framework.In three dimensional settings of practical interest, such systems are too large to be handleddirectly and must be solved iteratively. In this talk, I will cover the physical background ofthe problem class of interest, how such systems can be modeled, and then solved. Particularinterest will be paid to the radiation boundary conditions (perfectly matched layers versushigher order absorbing boundary conditions), issues associated with frequency domainversus time domain methods, and how these choices interact with iterative solvertechnologies in sometimes unexpected ways. Time permitting I will also touch upon the issue of harmonic inversion methods of this class of problems.

In this talk I will survey the results on the existence of solutions of the semigeostrophic system, a fully nonlinear reduction of the Navier-Stokes equation that constitute a valid model when the effect of rotation dominate the atmospheric flow. I will give an account of the theory developed since the pioneering work of Brenier in the early 90's, to more recent results obtained in a joint work with Mike Cullen and David Gilbert.

Let $M$ be the Cantor space or an $n$-dimensional manifold with $C(M,M)$ the set of continuous self-maps of $M$. We analyse the behaviour of the generic $f$ in $C(M,M)$ in terms of attractors and some notions of chaos.

We show the solvability of a proposed Generalized Buckley-LeverettSystem, which is related to multidimensional Muskat Problem. More-over, we discuss some important questions concerning singular limitsof the proposed model.

We study the mean-field models describing the evolution of distributions of particle radii obtained by taking the small volume fraction limit of the free boundary problem describing the micro phase separation of diblock copolymer melts, where micro phase separation consists of an ensemble of small balls of one component. In the dilute case, we identify all the steady states and show the convergence of solutions.

Next we study the dynamics for a free boundary problem in two dimension, obtained as a gradient flow of Ohta- Kawasaki free energy, in the case that one component is a distorted disk with a small volume fraction. We show the existence of solutions that a small, almost circular interface moves along a curve determined via a Green’s function of the domain. This talk is partly based on a joint work with Xiaofeng Ren.

Image segmentation and a number of other problems from image processing and computer vision can be regarded

as interface problems. Recently, diffusive and sharp interface techniques have been used for these problems.

In this talk, we will first briefly explain these models and compare the advantages and disadvantages of these models. Numerically, these models can be solved through some PDEs. In the end, we will show some recent results on how to use graph cut to solve these interface problems. Moreover, the global minimizer can be guaranteed even the problem is nonconex and nonlinear. The use of max-flow in a network setting and also in an infinite dimensional setting will be explained.

 Consider the solution of a scalar Helmholtz equation where the potential (or index) takes two positive values, one inside a disk or a ball (when d=2 or 3) of radius epsilon and another one outside. For this classical problem, it is possible to derive sharp explicit estimates of the size of the scattered field caused by this inhomogeneity, for any frequencies and any contrast. We will see that uniform estimates with respect to frequency and contrast do not tend to zero with epsilon, because of a quasi-resonance phenomenon. However, broadband estimates can be derived: uniform bounds for the scattered field for any contrast, and any frequencies outside of a set which tends to zero with epsilon.

In the talk I will mention two regularity results: the SBV regularity for strictly hyperbolic, genuinely nonlinear 1D systems of conservation laws and the characterization of intrinsic Lipschitz codimension 1 graphs in the Heisenberg groups. In both the contexts suitable scalar, 1D balance laws arise with very low regularity. I will in particular highlight the role of characteristics.

This seminar will be based on joint works with G. Alberti, S. Bianchini, F. Bigolin and F. Serra Cassano, and the main previous literature.