JHU Analysis and PDE Seminar


  Spring 2007

Mondays at 4:00pm in 308 Krieger Hall
Time
Speaker Title
Friday, January 26.
9:00-10:00am,
302 Krieger
Xiaoyi Zhang
(Chinese Academy of Sciences)		 Critical Nonlinear Schrödinger Equations
January 29

Hsungrow Chan
(National Pingtung University of Education, Taiwan) 		Nonpositively Curved Surfaces in R3
February 12

Ciprian Gal
(Morgan State University) 		Exponential Attractors for Singularly Perturbed Phase-Field Equations with Dynamic Boundary Conditions
February 26

Hamid Hezari
(Johns Hopkins) 		Complex Zeros of 1D Schrödinger Eigenfunctions
March 5

Atanas Stefanov
(University of Kansas) 		Pseudodifferential Operators with Rough Symbols and Applications
March 14-18
JAMI Conference Harmonic Analysis and Partial Differential Equations
Detailed information about speakers and activities is available from the Conference homepage.
March 19

Svetlana Roudenko
(Arizona State University) 		Concentration properties for the blow up solutions for the cubic NLS in 2 and 3D
March 26

Donatella Danielli
(Purdue University) 		Instability of Graphical Strips and a Positive Answer to the Bernstein Problem in the Heisenberg Group
April 2

Ryan Berndt
(Kansas State University) 		Recent Developments on the Two-Weight Problem for the Fourier Transform
Friday, April 6. 2:00-3:00pm
304 Krieger
Ovidiu Savin
(Columbia University) 		On Monge-Ampere Equations with Homogeneous Right Hand Side
April 16

Hans Christianson
(University of California, Berkeley) 		Dispersive Equations and Hyperbolic Orbits
April 23

Edward Saff
(Vanderbilt University) 		Discrete Minimal Energy Problems

Analysis Seminar Calendar for Past Semesters


Abstract of Talks:

January 26: Xiaoyi Zhang (Chinese Academy of Sciences):   Critical Nonlinear Schrödinger Equations.

January 29: Hsungrow Chan (National Pingtung University):   Nonpositively Curved Surfaces in R3.

In this talk we discuss generalizations of some minimal surface theorems in Euclidean three space to surfaces with nonpositive Gauss curvature and square integrable second fundamental form. For example we show that if a complete simply connected nonpositively curved surface that is twice continuously differentiably embedded in Euclidean three space has square integrable second fundamental form, then it must be a plane. Further, if a complete nonpositively curved surface with one end is embedded near infinity and has square integrable second fundamental form, then it must lie between two parallel planes. These results may be considered generalizations of Bernstein's minimal surface theorems.


February 12: Ciprian Gal (Morgan State University):   Exponential Attractors for Singularly Perturbed Phase-Field Equations with Dynamic Boundary Conditions.

We construct a robust family of exponential attractors for singularly perturbed phase-field type equations with dynamic boundary conditions for the order parameter function. Moreover, we obtain estimates for the symmetric distance between these exponential attractors and that of the limit Cahn-Hilliard equation with dynamic boundary conditions, in terms of the perturbation parameter. This is joint work with Maurizio Graselli.


February 26: Hamid Hezari (Johns Hopkins):   Complex Zeros of 1D Schrödinger Eigenfunctions.

I will talk about the distribution of the complex zeros of 1D Schrödinger eigenfunctions as h®0 for the class of polynomial potentials. I will show that the zeros tend to concentrate on some lines in the complex plane as h®0. I will also illustrate these lines for some double-well potentials.


March 5: Atanas Stefanov (University of Kansas):   Pseudodifferential Operators with Rough Symbols and Applications.

Classical estimates (like Hörmander-Mikhlin, Calderon-Vaillancourt, Coifmann-Meyer) for pseudodifferential operators (PDO) require pointwise bounds on certain number of derivatives in both the x and ξ variables. In the applications however, the PDO that arise rarely satisfy such stringent assumptions.

In this talk, I will present some recent results for Lp boundedness of PDO, which require only boundedness in the x variable and Sobolev type condition (i.e. L2 integrability of certain derivatives) on the ξ variable. The results are sharp up to an endpoint. It turns out that one can even study maximal singular integral operators in this framework. As an application and with simple proofs consisting of essentially checking the conditions, I will show the Lp bounds for (a slightly smoothed out version of) the Carleson-Hunt operator and appropriate Lp bounds for maximal operator closely related to the Kakeya maximal function.


March 19: Svetlana Roudenko (Arizona State University):   Concentration properties for the blow up solutions for the cubic NLS in 2 and 3D.

First, I will discuss Bourgain's mass concentration phenomenon for the cubic nonlinear Schrodinger (NLS) equation in 2D. I will focus on the refinement of this technique which links the divergence of Strichartz norms with the size of the concentration window. As a consequence, a logarithmic lower bound on the blow up rate of the relevant Strichartz norm will be obtained. Next, I will consider the focusing cubic NLS in 3D and discuss the L3 concentration phenomenon and consequences related to the dynamics of blow up solutions.


March 26: Donatella Danielli (Purdue University):   Instability of Graphical Strips and a Positive Answer to the Bernstein Problem in the Heisenberg Group.


April 2: Ryan Berndt (Kansas State University):   Recent Developments on the Two-Weight Problem for the Fourier Transform.

I will talk about some very recent developments concerning the problem of finding conditions on weights so that the Fourier transform maps one weighted Lp space into a different weighted Lp space.


April 6: Ovidiu Savin (Columbia University):   On Monge-Ampere Equations with Homogeneous Right Hand Side.

We consider the degenerate Monge Ampere equation  det D2u(x)=|x|α   in 2D and discuss the behavior at the origin and the regularity of u. This a joint work with P. Daskalopoulos.


April 16: Hans Christianson (UC Berkeley):   Dispersive Equations and Hyperbolic Orbits.

It is well known that on compact manifolds, classical ergodicity leads to uniform distribution of eigenfunctions, and classical control assumptions lead to good resolvent estimates. For non-compact manifolds, non-trapping assumptions yield good resolvent estimates. In both compact and non-compact cases, we show the presence of a closed hyperbolic orbit causes a logarithmic loss in the resolvent estimates. This is a very small loss, which follows from the unstable nature of hyperbolic orbits.


April 23: Edward Saff (Vanderbilt University):   Discrete Minimal Energy Problems.

For a compact set A in Euclidean space we shall investigate the asymptotic behavior of optimal (and near optimal) N-point configurations that minimize the Riesz s-energy (corresponding to the potential 1/r^s for s>0 and log(1/r) for s=0) over all N-point subsets of A, where r denotes Euclidean distance. If A has finite and positive d-dimensional Hausdorff measure and s < d, then the analysis of such points falls under the umbrella of classical potential theory and is a consequence of the continuous theory.

But what if s > d or s=d? In such cases, the classical theory does not apply since A has s-capacity zero and so new techniques are needed to analyze the behavior of minimal energy configurations. We shall describe these techniques, which also yield information about "best-packing points" on A; that is, N points of A for which the minimal pairwise distance is as large as possible.

The talk represents joint work with Doug Hardin and Sergiy Borodachov.

References:

1)D.P. Hardin and E.B. Saff, Discretizing Manifolds via Minimum Energy Points, Notices of the American Mathematics Society, November 2004, pp.1186-1194.

2)D.P. Hardin and E.B. Saff, Minimal Riesz Energy Point Configurations for Rectifiable d-Dimensional Manifolds, Advances in Mathematics, Vol. 193, No. 1 (2005), pp. 174-204.

3)S.V. Borodachov, D.P. Hardin and E.B. Saff Asymptotics for Discrete Weighted Minimal Riesz Energy Problems on Rectifiable Sets, Trans. Amer. Math. Soc. (to appear)

4)D. Hardin, E.B. Saff and H. Stahl, The Support of the Logarithmic Equilibrium Measure on Sets of Revolution in R3-, will appear in J. Math. Physics.

5) S. Borodachov, D.P. Hardin, and E.B. Saff, Asymptotics of Best-Packing on Rectifiable Sets, to appear in Proc. Amer. Math. Soc.

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