## Department of Mathematics, University of Kentucky PDE Seminar

Spring 2009

Date SPEAKER and TITLE Host
January 20 Professor Eric Soccorsi, CPT CNRS Luminy, Marseille, France
TITLE:Eigenvalue asymptotics for a twisted waveguide
ABSTRACT : pdf file
Peter Hislop
January 27 Mr. Justin Taylor, University of Kentucky
TITLE: The Dirichlet eigenvalue problem for the Lame system and for elliptic system on perturbed domains
ABSTRACT: pdf file
Russell Brown
February 3 Dr. Tomasz Adamowicz, University of Cincinnatti
TITLE: On the geometry of the p-harmonic world
ABSTRACT : pdf file
John Lewis
February 10 Professor Min-Chun Hong, University of Queensland, Australia
TITLE: Global Existence of the Siberg-Witten Flow
ABSTRACT : pdf file
Changyou Wang
February 13 Professor Andrew Lorent, University of Cincinnatti
TITLE: Quantitative Liouville Theorems
ABSTRACT: pdf file
John Lewis
February 17 Professor Vladimir Eyderman, University of Kentucky
TITLE: Cartan Type Estimates on Riesz Transform
ABSTRACT : Our aim is to give sharp upper bounds for the size of the set of points where the singular Riesz transform of a linear combination of N point masses is large. This size will be measured by the Hausdorff content with various gauge functions. Among other things, we shall characterize all gauge functions for which the estimates do not blow up as N tends to infinity (in this case a routine limiting argument will allow us to extend our bounds to all finite Borel measures). This is a joint work with F.Nazarov and A.Volberg
James Brennan
February 24 Professor Luca Capogna, University of Arkansas
TITLE: Regularity of certain minimal graphs in the sub- Riemannian Heisenberg group
ABSTRACT: Minimal surfaces in the sub-Riemannian Heisenberg group can be constructed by means of a Riemannian approximation scheme, as limit of Riemannian minimal surfaces. We study the regularity of Lipschitz, non-characteristic minimal surfaces which arise as such limits. Our main results (joint with Citti and Manfredini) are a-priori estimates on the solutions of the approximating Riemannian PDE and the ensuing $C^ {\infty}$ regularity of the sub-Riemannian minimal surface along its Legendrian foliation.
Zhongwei Shen
March 3 Professor Yuan, Yu, University of Washington/IAS (Postponed)
TITLE: tba
ABSTRACT :
Changyou Wang
March 10 Professor David Adams, University of Kentucky (Cancelled)
TITLE: Wolff potentials and solutions to the inhomogeneus wave equation in 3 space dimensions
ABSTRACT: tba
Changyou Wang
March 12
(Colloquium)
Professor Vladimir Sverak, University of Minnesota
TITLE: tba
ABSTRACT : tba
Changyou Wang
March 17 No seminar, Spring break week
TITLE:
ABSTRACT :
TBA
March 24 No seminar
TITLE : tba
ABSTRACT : tba
March 31 Professor Peter Perry, University of Kentucky
TITLE : Inverse Scattering as Nonlinear Fourier Analysis
ABSTRACT : A number of nonlinear dispersive equations such as the Korteweg-de Vries Equation (KdV) $u_t + u_xxx+ 6uu_x =0$ and the nonlinear Schrodinger equation (NLS) $i u+t + u_xx+|u|^2 u = 0$ (both for a function u of one space and one time variable) are solvable by the inverse scattering method. In this talk, I will briefly discuss the role these equations play in the theory of nonlinear dispersive waves, then discuss the inverse scattering method for solving the NLS in detail. In summary, there is an invertible nonlinear map $\mathcal{R}$ which maps solutions of the nonlinear equations to solutions of the same equation but with the nonlinearity removed. We'll construct the mapping $\mathcal{R}$ and discuss how it behaves like a "nonlinear Fourier transform". This is a partly expository talk and will allude both to "classical" work in inverse scattering from the '60's and '70's, and also to more recent work by Tao and collaborators. It is also a shameless plug for Math 773, Topics in Analysis - The Nonlinear Schrodinger Equation, to be offered in Fall 2009.
Changyou Wang
April 7 Professor Richard Laugesen, UIUC (Postponed)
TITLE :
ABSTRACT :
Peter Hislop
April 14 Professor Olaf Post, Humboldt University, Germany
TITLE : Quantum graph approximations of thin branching structures
ABSTRACT : Many physical systems have branching structure of thin transversal diameter. One can name for instance quantum wire circuits, thin branching waveguides, or carbon nano-structures. In applications, such systems are often approximated by the underlying one-dimensional graph structure, a so-called "quantum graph". In this way, many properties of the system like conductance can be calculated easier (sometimes even explicitly). After briefly explaining the notion of a quantum graph, we show that the system with thin transversal diameter converges to a quantum graph. We also identify which vertex couplings (which influence the current through the graph) can be obtained by appropriate engineering of the branching structure.
Peter Hislop
April 16
(Hayden-Howard Lecture)
Professor Carlos Kenig, University of Chicago
TITLE: The global behavior of solutions to critical nonlinear dispersive and wave equations
ABSTRACT : see department handout
Zhongwei Shen
April 21 Dr. Mikko Parviainen, Helsinki Institute of Technology
TITLE: Nonlinear PDEs and stochastic games
ABSTRACT : The theory of partial differential equations is closely related to stochastics. For example, the links between the Laplace operator and Brownian motion are well known. Recently, Peres, Schramm, Sheffield, and Wilson showed that a random turn tug-of-war game approximates the infinity harmonic functions. In addition, Peres and Sheffield studied a connection between the p-harmonic functions and the random turn tug-of-war with noise. We study the nonlinear PDEs and stochastic games in a context of asymptotic expansions. This talk is based on a joint work with J.J. Manfredi and J.D. Rossi.
John Lewis
April 23
(Colloquium)
Professor Yuan, Yu, University of Washington/IAS
TITLE: Recent results for Special Lagrangian equations
ABSTRACT : We survey some recent results on Hessian, gradient estimates, regularity, and global rigidity for special Lagrangian equations with certain convexity. The gradient graphs of the solutions are minimal Lagrangian submanifolds in Euclidean space. The special Lagrangian equations in the Pseudo-Euclidean setting are just Monge-Ampere equations, for which one has the corresponding classic positive results and counterexamples. Part of the work is joint with Warren, some also with Chen.
Changyou Wang
April 28 Professor Peter Topalov, Northeastern University
TITLE: On the Integral Geometry of Liouville Billiard Tables
ABSTRACT : A notion of Radon transform for completely integrable billiard tables is introduced. It will be shown that in the case of Liouville billiard tables of dimension 3 the Radon transform is one-to-one on the space of continuous functions $K$ on the boundary of the billiard. This allows us to obtain spectral rigidity of Laplace-Beltrami operator with Robin boundary conditions in certain domains.
Peter Perry
May 5 Professor Richard Laugesen, UIUC
TITLE : Low eigenvalues of the Neumann Laplacian on triangles
ABSTRACT : Szego and Weinberger showed that the fundamental tone of a free membrane is maximal for the ball, among regions of given volume. That is, the first nonzero eigenvalue of the Laplacian under Neumann boundary conditions is maximal for the ball. In the opposite direction, Payne and Weinberger showed the fundamental tone is minimal for the degenerate rectangle, among convex regions of given diameter. We tackle analogous extremal problems for triangles. Neither Szego's conformal maps nor Weinberger's extended trial functions nor Payne and Weinberger's slicing arguments work for triangular domains. Instead we develop linear transplantation, and the "method of the unknown trial function". [Joint work with B. Siudeja, Univ. of Illinois]
Peter Hislop
May 12
(Special PDE Seminar)
Professor Lin,Tai-Chia National University of Taiwan/IMA
TITLE: Skyrmions in Gross-Pitaevskii functionals
ABSTRACT : Recently, Skyrmions with integer topological charges have been observed numerically but have not yet been shown rigorously on two-component systems of Gross-Pitaevskii equations (GPEs) describing a binary mixture of Bose-Einstein condensates (BEC). Here we construct skyrmions by studying critical points of Gross-Pitaevskii functionals with two-component wave functions. Using localized energy method, we rigorously prove the existence and configuration of skyrmions in BEC. On the other hand, half-Skyrmions characterized by half-integer topological charges can also be found in the nonlinear sigma model which is a model of BEC of the Schwinger bosons. Here we also prove rigorously the existence of half-Skyrmions which may come from a new type of soliton solutions called spike-vortex solutions of two-component systems of GPEs.
Changyou Wang