JHU - Analysis and PDE Seminar

JHU Analysis and PDE Seminar Spring 2008 Mondays at 3:00pm in 308 Krieger Hall
Date	Speaker	Title
February 4	Peter Ebenfelt (UC San Diego)	Super-Rigidity of CR Embeddings into Pseudo-Concave Hyperquadrics
February 11	Jeremy Marzuola (Columbia University)	Stability of Minimal Mass Soliton Solutions for Saturated Nonlinear Schrödinger Equations
February 18	Michael Frazier (University of Tennessee)	Estimates for Green's Functions of Schrödinger Operators
March 3	Ramona Anton (Université Paris Sud XI, visiting JHU)	Non-Linear Schrödinger Equations on Domains with Boundary
March 10	Jason Metcalfe (University of North Carolina)	Strichartz Estimates for Wave Equations on Schwarzschild Black Hole Backgrounds
March 17	Spring Break	No Seminar This Week
March 24	Johan Martens (University of Toronto)	The Hitchin System for Parabolic Higgs Bundles
March 31	Nikolaos Tzirakis (University of Illinois)	Correlation Estimates and Applications to Nonlinear Schrödinger equations
Tues., April 15	Matt Blair (University of Rochester)	Strichartz Estimates for the Wave Equation on Domains and Applications
April 21	Henghui Zou (University of Alabama, Birmingham, visiting JHU)	A Priori Estimates and Existence for Quasi-Linear Elliptic Equations
Wed., April 23 4:00-5:00pm Mergenthaler 111	Craig Evans (UC Berkeley)	SIMCA Colloquium: A Nonlinear PDE Model for Lakes and Rivers
April 28	Qing Han (University of Notre Dame)	Isometric Embedding of Compact Surfaces in R³
Wed., May 7 1:00-2:00pm Krieger 302	Mohammad Ghomi (Georgia Tech)	Topology of Riemannian Submanifolds with Prescribed Boundary
May 12	Mikko Salo (University of Helsinki)	Carleman Estimates and Calderón's Inverse Problem on Manifolds

Analysis Seminar Calendar for Past Semesters

Abstract of Talks:

February 4: Peter Ebenfelt (UCSD): Super-Rigidity of CR Embeddings into Pseudo-Concave Hyperquadrics.

In the '70s, Alexander discovered an interesting rigidity phenomenon for holomorphic mappings sending a piece of the unit sphere in Cⁿ⁺¹, n ≥ 1 , into itself: Such a map is either constant or extends as a linear fractional mapping sending the ball into itself. This was later generalized, by numerous mathematicians, to mappings sending the sphere in Cⁿ⁺¹ into that in C^N+1 provided that N-n < n, and the latter codimensional estimate is sharp. It was recently discovered by Baouendi and Huang that, surprisingly, for mappings between pseudo-concave hyperquadrics, this rigidity phenomenon holds without any restriction on the codimension N-n. We will discuss some generalizations of these type of results to the case where the source manifold is not necessarily a quadric.

February 11: Jeremy Marzuola (Columbia): Stability of Minimal Mass Soliton Solutions for Saturated Nonlinear Schrödinger Equations.

In this result, we develop the techniques of Schlag, Krieger-Schlag and Bourgain-Wang in order to determine a class of stable perturbations for a minimal mass soliton solution of a saturated, focusing nonlinear Schrödinger equation (NLSE) in R³. We first numerically and anaytically study the existence of a minimal mass soliton, as well as the spectrum of the Hamiltonian resulting from linearizing about that soliton. By projecting into a subspace of the continuous spectrum of the linearized Hamiltonian, we are able to use a contraction mapping in order to show that there exist solutions which look like the soliton plus linear dispersion.

February 18: Michael Frazier (Tennessee): Estimates for Green's Functions of Schrödinger Operators.

If T is a bounded linear operator on L²(μ) with norm less than one, then (I-T) has an inverse given by a Neumann series. Suppose T is represented by integration against a symmetric kernel K(x,y). Under the condition that the reciprocal of K is a quasimetric, we obtain an exponential lower bound for the kernel of the inverse of I-T. Under an appropriate smallness condition on T, we obtain an upper bound of the same type.

These results were motivated by the inhomogeneous, time-independent Schrodinger equation. We obtain estimates for the Green's function of the Schrodinger operator for a very general class of domains. Examples include the potential -c|x|^-2 in n dimensions for n>2.

Our methods also apply to operators with fractional potential replacing the Laplacian. These operators relate to alpha-stable Levy processes in the same way that the Laplacian relates to Brownian motion.

March 3: Ramona Anton (Orsay/JHU): Non-Linear Schrödinger Equations on Domains with Boundary.

We are interested in proving global existence results in the energy space for the semi-linear Schrödinger equation on domains of dimension 2 or 3. The main ingredients are generalized Strichartz inequalities adapted to the domains, which have some loss of derivatives. We present the results and the strategy for three types of domains. On exterior domains of dimension three the corresponding inequality enables us to study also the Cauchy problem for the Gross-Pitaevskii equation.

March 10: Jason Metcalfe (UNC): Strichartz Estimates for Wave Equations on Schwarzschild Black Hole Backgrounds.

We prove Strichartz estimates for wave equations on Schwarzschild black hole backgrounds. This is done by combining some local energy estimates with a global-in-time outgoing parametrix for small perturbations of the d'Alembertian. Particular care needs to be taken near the regions of trapping, namely the event horizon and the photon sphere. This is a joint work with J. Marzuola, D. Tataru, and M. Tohaneanu.

March 24: Johan Martens (Toronto): The Hitchin System for Parabolic Higgs Bundles.

Introduced in 1987, the Hitchin system on the moduli space of Higgs bundles is one of the prime examples of an algebraically completely integrable system. It exhibits an extraordinary rich geometric structure, that can be studied from different geometric angles - differential, algebraic or hyperkaehler - and ties in beautifully with representation theory, the geometric langlands program, and physics. In this talk we discuss work in progress with M. Logares on a variation thereon, the Hitchin system for (non-strongly) parabolic Higgs bundles.

March 31: Nikolaos Tzirakis (Illinois): Correlation Estimates and Applications to Nonlinear Schrödinger equations.

In this talk I will show how one can obtain new interaction Morawetz type (correlation) estimates in one and two dimensions. These estimates correspond to the nonlinear diagonal analogue of Bourgain's bilinear refinement of Strichartz. The framework is quite general and we can simultaneously obtain the global a priori estimates that have been obtained before in dimensions higher than three. In higher dimensions the proof uses commutator vector operators acting on the conservation laws of the equation. In one dimension we use the Gauss-Weierstrass summability method acting on the conservation laws. I will then show several applications of the new estimates to the semilinear Schrödinger equation.

April 15: Matt Blair (Rochester): Strichartz Estimates for the Wave Equation on Domains and Applications.

Strichartz estimates are a family of space-time integrability estimates for the wave equation that rely on the dispersive effect of the solution map. Such inequalities are well-established when the problem is posed over Euclidean space as well as in other contexts. However, when boundary conditions are present, much less is known. We discuss new results in the area, and applications to certain semilinear wave equations. This is a joint work with H. Smith and C. Sogge.

April 21: Henghui Zou (UAB): A Priori Estimates and Existence for Quasi-Linear Elliptic Equations.

We study the boundary value problem of quasi-linear elliptic equation
div(|∇ u|^m-2∇ u) + B(z,u,∇ u) = 0 in Ω,
u = 0 on ∂Ω
where Ω⊂Rⁿ (n≥2) is a connected smooth domain, and the exponent m ∈ (1,n) is a positive number. Under appropriate conditions on the function B, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established. The results are generically optimum for the canonical prototype B=|u|^p-1u, p>m-1.

April 23: Craig Evans (Berkeley): A Nonlinear PDE Model for Lakes and Rivers.

I will briefly discuss some simple (and not-so-simple) nonlinear PDEs describing growing "sandpiles". I will then introduce a new nonlinear PDE that in an asymptotic limit models the formation of "lakes" and "rivers" resulting from rainfall over a fixed landscape.

These toy equations illustrate the serious point that interesting phenomena often appear when we let the parameters in PDEs approach infinity.

April 28: Qing Han (Notre Dame): Isometric Embedding of Compact Surfaces in R³.

In early 1950s, Nirenberg and Pogorelov independently gave an affirmative answer to the Weyl problem: Any smooth metric on S² with positive Gauss curvature admits a smooth isometric embedding in R³. This is the only result concerning the isometric embedding of compact surfaces in R³. In this talk, we will discuss the isometric embedding of other compact surfaces in R³. When Gauss curvature changes sign, fundamental equations for isometric embedding are of mixed type, elliptic when Gauss curvature is positive and hyperbolic when Gauss curvature is negative. Very little is known for global solutions of differential equations of mixed type. Our strategy to construct isometric embedding is to solve these equations first where Gauss curvature is positive and then extend solutions to regions where Gauss curvature is negative. An important step is to ensure that a closed surface is formed. Rigidity of compact surfaces will also be discussed.

May 7: Mohammad Ghomi (Georgia Tech): Topology of Riemannian Submanifolds with Prescribed Boundary.

We prove that a smooth compact submanifold of codimension 2 immersed in Rⁿ, n>2, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. This settles a question of Guan and Spruck related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimension is too high, or the prescribed boundary is not sufficiently regular. Our proofs employ, among other methods, a relative version of Nash's isometric embedding theorem, and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman. These results consist of joint works with Stephanie Alexander and Jeremy Wong, and Robert Greene.

May 12: Mikko Salo (Helsinki): Carleman Estimates and Calderón's Inverse Problem on Manifolds.

We consider the imaging of anisotropic materials by electrical measurements. This inverse problem arises in Electrical Impedance Tomography (EIT), which has been proposed as a diagnostic method in medical imaging and nondestructive testing. The mathematical model is the anisotropic Calderón problem, which consists in determining a matrix of coefficients in an elliptic equation from boundary measurements of solutions.

In geometric terms, the problem is to determine a Riemannian metric from Cauchy data of harmonic functions on a manifold. Our approach is based on Carleman estimates. We characterize those Riemannian manifolds which admit a special limiting Carleman weight. By using these weights we construct complex geometrical optics solutions to elliptic equations, and prove uniqueness results in inverse problems for a class of Riemannian manifolds.

This is a joint work with D. Dos Santos Ferreira (Paris 13), C. Kenig (Chicago), and G. Uhlmann (Washington).

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Analysis Seminar Calendar for Past Semesters

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JHU Analysis and PDE Seminar

Mondays at 3:00pm in 308 Krieger Hall