Seminars in Analysis and PDEs

Unless otherwise stated, Analysis and PDE Seminars are held at 2pm on Tuesday in the Bernhard Neumann Seminar Room (G35) on the ground floor of the John Dedman Mathematical Sciences Building, Bldg 27 (Map).
Contact Julie Clutterbuck for information.

View MSI weekly bulletin (includes all seminar and colloquium details for the week).

Second Semester 2008

Date	Speaker	Affiliation	Title/Abstract
21/10/2008	Andrew Hassell	ANU	Restriction theorems on asymptotically conic manifolds Abstract: This is a report on ongoing research with Colin Guillarmou and Adam Sikora. The Stein-Tomas restriction theorem on $n$-dimensional Euclidean space says that the Fourier transform of an $L^p$ function restricts boundedly to an $L^2$ function on the unit sphere for $p$ between 1 and $2(n+1)/(n+3)$; equivalently, the spectral measure $dE(\lambda)$ for the Euclidean Laplace operator is a bounded operator between $L^p$ and its dual space $L^{p'}$ for $p$ in the same range. We ask the same question on asymptotically conic manifolds $M$; that is, for which values of $p$ is the spectral measure $dE(\lambda)$ for the Laplacian on $M$ a bounded operator between $L^p(M)$ and its dual space $L^{p'}(M)$? Under some geometric assumptions concerning conjugate points on the manifold, we prove boundedness for the same range of $p$ as on Euclidean space. Moreover, it appears that our geometric assumptions are essentially sharp.
21/11/2008 2pm, G35	Min Ji	Academy of Mathematics and Systems Science, Chinese Academy of Sciences	Scalar Curvature Equation on Sⁿ Abstract: This is devoted to the study of the prescribing scalar curvature problem on the standard sphere of any dimension.
9/12/2008	Andrew Morris	ANU	Local Hardy Spaces of Differential Forms Abstract: The Riesz transforms $\partial_j\Delta^{-\frac{1}{2}}$, where $\Delta=-\sum_{j=1}^n \partial_j{}^2$ is the Laplacian, are bounded on $L^p(\mathbb{R}^n)$ for $p\in(1,\infty)$. At the endpoints one subs titutes the Hardy space $H^1(\mathbb{R}^n)$ and its dual $BMO(\mathbb{R}^n)$ to recover boundedess. The geometric Riesz transform $D\Delta^{-\frac{1}{2}}$, where $D=d+d^*$ is the Hodge-Dirac operator and $\Delta=D^2$ is the Hodge-Laplacian, is form-valued. This lead Auscher, McIntosh and Russ to def ine Hardy spaces of differential forms on doubling Riemannian manifolds. The boundedness of the geom etric Riesz transform is essentially built-in to their definition. We will survey these ideas and recent progress towards developing a corresponding theory for local H ardy spaces on manifolds where the metric-measure interaction is allowed to be locally exponentially doubling. In these spaces the boundedness of the local Riesz transform $D(\Delta+a)^{-\frac{1}{2}}$ is built-in provided that $a$ is large enough in comparison with the exponential growth of the mani fold. In particular, the exponential off-diagonal estimates and some new function spaces that we hav e used will be discussed.
Wednesday 14/1/2009 2pm, G35	Sevvandi Kandanaarachchi	Monash University	Easy proof for convex surfaces flowing into spheres by Mean Curvature Flow
27/1/2009	Tom ter Elst	University of Auckland	Sectorial forms
17/2/2009	Alex Barnett	Dartmouth College	TBA
24/2/2009	Todd Oliynyk	Monash University	Post-Newtonian expansions for the cosmological Einstein-Euler equations Einstein's general relativity is presently the most accurate theory of gravity. To completely determine the gravitational field, the Einstein field equations must be solved. These equations are extremely complex and outside of a small set of idealized situations they are impossible to solve directly. However, to make physical predictions or understand physical phenomena, it is often enough to find approximate solutions that are governed by a simpler set of equations. The prime example of this is Newtonian gravity which approximates general relativity very well in regimes where the typical velocity of the gravitating matter is small compared to the speed of light. Indeed, Newtonian gravity successfully explains much of the behavior of our solar system and is a simpler theory of gravity that is less difficult to solve. By generalizing Newtonian gravity to the cosmological setting, it appears that Newtonian theory can accurately describe gravity on all scales except in regions of very high curvature such as near compact neutron stars or black holes. However, for many situations of interest ranging from binary star systems to GPS satellites, the Newtonian approximation is not accurate enough; general relativistic effects must be included. The desire to include relativistic corrections to Newtonian gravity then lead to the development of the post-Newtonian expansions. The goal of the post-Newtonian expansions is to approximate solutions to the Einstein field equations by a series expansion in the parameter v/c where c is the speed of light and v is a typical speed associated with the gravitating matter. The difficulty with rigorously understanding the post-Newtonian expansions is that the region of validity for the expansions is where v/c is close to zero. Consequently, solutions of general relativity must be examined in the limit that v/c -> 0 and in this limit the Einstein field equations become singular; that is the field equations contains terms of the form c/v that become unbounded as v/c -> 0. In this talk, I will describe recent work that rigorously justifies the post-Newtonian expansions for the cosmological Einstein-Euler equations which govern a gravitating perfect fluid. More specifically, I will describe how the problem of generating post-Newtonian expansions to a certain order can be reduced to solving a system of non-linear elliptic equations to determine initial data along with energy estimates for a particular non-local symmetric hyperbolic formulation of the Einstein-Euler equations. Time permitting; I will also describe related, but less complete, results for the Einstein-Euler equations that are relevant for isolated systems.
17/3/2009	Shi-jun Zheng	Georgia Southern University
24/3/2009	Uta Freiberg	University of Jena	Pruefer angle methods in spectral analysis on the real line Generalized second order differential operators of the form d/dm d/dx are considered. They act on the space L_2([0,1], m), where m is an atomless measure which is in general singular with respect to the Lebesgue measure. In the particular case that m is self-similar, one obtains Weyl asymptotics of the eigenvalues which can be refined by applying renewal theory. In some special cases, the method of Pruefer angles leads to exact renormalization formulas for the Neumann eigenvalues, allowing a better study of the spectral asymptotics in the lattice case.
21/4/2009	Robert Taggart	ANU	Stricharz estimates Strichartz estimates are the basic estimates for solutions to linear wave and Schrodinger equations. They can be used to prove the well-posedness results in a nonlinear setting and are also equivalent to various restriction theorems for the Fourier transform. While all Strichartz-type estimates for the homogeneous wave and Schrodinger equations in Euclidean space are known, the question of finding all Strichartz estimates for the corresponding inhomogeneous equations remains open. In this talk, I will give an historical survey of this field, including the work of Robert Strichartz (1970 and 1977), Yajima (1980s), Ginibre--Velo (1995) and the influential paper of Keel--Tao (1998).I will then look at recent progress, including that by the speaker, in determining all Strichartz estimates for the inhomogeneous wave, Schrodinger and related equations.
28/4/2009	Andreas Axelsson	Stockholm University	Semigroup methods for elliptic non-smooth systems of PDEs Semigroup/group methods are standard tools for finding the time evolution of solutions to linear parabolic/hyperbolic equations like the heat/wave equation. Less well understood is how to use semigroup methods to solve elliptic boundary value problems (BVPs), a reason being that the operator that plays the role of an infinitesimal generator has a two-sided unbounded spectrum. For elliptic equations, the evolution variable $t$ is the coordinate transversal to the boundary rather than time. In this talk I will discuss joint work with Alan McIntosh and Pascal Auscher, where we use semigroup methods and harmonic analysis techniques from the proof of the Kato square root estimate to solve BVPs for elliptic systems of PDEs with general non smooth and complex coefficients. The strength of our methods is that they apply not only to scalar equations but to quite general systems of PDEs, in particular to second order divergence form systems. I will also report on work in progress with P. Auscher, which takes these methods beyond semigroups and $t$-independent coefficients.
12/5/2009	Valentina Vulcanov	Freie Universität Berlin	Mean curvature of graphs with free boundaries We treat a particular case of mean curvature flow of graphs with mixed boundary conditions, as a continuation of work done by A. Stahl and J. Buckland. We study the case of radially symmetric graphs evolving outside a sphere with moving Neumann boundary condition on the sphere and a Dirichlet boundary condition at a fixed radius outside the sphere. In the case of long time existence we show that the solution converges to a constant function and give an overview of how the symmetric case can be used to investigate long time existence for general graphs with the same boundary conditions.
19/5/2009	Melissa Tacy	ANU	Semiclassical L^p Estimates of Quasimodes on Submanifolds In this talk I will present L^p estimates for restrictions of approximate eigenfunctions (quasimodes) to submanfolds. This work uses properties of the flow associated with the symbol of an operator to convert the problem into one concerning evolution equations. Strichartz estimates are then used to obtain the L^p estimates.
9/6/2009	Carlo Nitsch	Naples
23/6/2009	Markus Hegland	ANU	A new type of interpolation inequalities based on dilation and their application in spectral narrowing Interpolation inequalities are an essential tool to analyse convergence of regularisation techniques for the solution of linear ill-posed problems. Using spectral theory of unbounded real symmetric linear operators, variable Hilbert scales were introduced for this task in the past. I will review this earlier work and present some results for a type of Hilbert scales which are based on dilation. Finally, I will discuss how these Hilbert scales can be used analyse some algorithms which enhance the resolution of measured spectra in near-infrared spectroscopy for example.
30/7/2009	Andreas Vasy	Stanford	Wave propagation on asymptotically De Sitter and De Sitter-Schwarzschild spaces In this project, which is partly joint work with Richard Melrose and Antonio Sa Barreto, we study the asymptotics of solutions of the wave equation. In the case of De Sitter-Schwarzschild space this is achieved by constructing a high energy parametrix for the analytic continuation of the resolvent of the Laplacian on asymptotically hyperbolic spaces; this parametrix is valid uniformly in a strip (in the complex plane of the spectral parameter) beyond the real axis. We use this and cutoff resolvent estimates of Bony and Haefner to obtain resolvent estimates on weighted spaces for the spatial "Laplacian" on De Sitter-Schwarzschild space (which, near infinity, is close to the hyperbolic Laplacian).Some similar results on wave asymptotics were obtained by Dafermos and Rodinianski.
7/7/2009	Tuomas Hytönen	Helsinki	Pseudo-localization of singular integrals in L^p As an intermediate step in the development of a noncommutative Calderon-Zygmund theory, J. Parcet (JFA, 2009) established a new ''pseudo-localization principle'' of classical singular integral operators in L² . For a given function f in L², this interesting principle provides a set outside of which any normalized singular integral of f has small norm. I will discuss this result and its extension to the reflexive L^p spaces, which was left open in Parcet's work. The new method of proof, based on martingale techniques, even slightly sharpens the original L² result.
14/7/2009			No seminar this week
21/7/2009	Dorina Mitrea	Missouri	On the regularity of the Dirichlet Green potential in convex domains In this talk I will discuss mapping properties of the Dirichlet Green potential on the scale of Besov and Triebel-Lizorkin spaces when the underlying domain satisfies a uniform exterior ball condition or, it is a convex domain. In the process, the Dirichlet and Regularity problems for harmonic functions in convex domains, with optimal nontangential maximal function estimates, will be treated.
Wednesday 22/7/2009 G35, 3pm	Jingye Chen	University of British Columbia	Curvature flow for Lagangian graphs Mean curvature flow is a curvature flow along which submanifolds evolve in the direction of their mean curvature. We will discuss recent results on longtime existence, uniqueness, and self-similar solutions for mean curvature flow of Lagrangian graphs.
28/7/2009	Dennis Labutin	UC Santa Barbara	Partial regularity for Monge-Ampere equation
25/8/2009	Feng Xu	ANU	Laplacian Flow A G₂-structure on a 7-dimensional manifold M is given by a definite 3-form &sigma . Such a G₂-structure is called closed if &sigma is closed, and torsion-free if &sigma is both closed and co-closed. The Riemannian holonomy of a torsion-free G₂-structure is contained in the Lie group G₂. In particular, the underlying metric will be Ricci-flat. The Laplacian flow deforms the G₂-structure along its Hodge Laplacian. In this talk, I will review G₂-structures and G₂-holonomy, and introduce Laplacian flow. After this, I will present joint work with R. Bryant on short-time existence of the Laplacian flow in the category of smooth closed G₂-structures. Then I will talk about joint work with Rugang Ye on dynamic stability of torsion-free G₂-structures under the Laplacian flow.
8/9/2009	Robert Taggert	ANU
22/9/2009			No seminar this week

20/10/2009	Rod Gover		Extending the Poincare-Einstein programme and overdetermined PDE A compact manifold with boundary is said to have a Poincare-Einstein structure if its interior is equipped with a negative curvature Einstein metric, in terms of which the boundary is suitably ``at infinity'' and has induced on it a conformal structure. A central problem is to relate the conformal geometry of this boundary to the Riemannian structure of the interior, and this is linked to the ideas behind Maldacena's AdS/CFT correspondence in String theory. There is a natural approach to Poincare-Einstein manifolds via conformal geometry and a certain overdetermined PDE. This leads to an elegant and effective way to treat many aspects of these structures. It also suggests natural ways to extend the programme and new problems in geometric analysis.
27/10/2009	Ben Hambly	Oxford	Heat kernels and local limit theorems for random walks on graphs The classical local limit theorem gives the convergence of the heat kernel for random walk on Z^d to the Gaussian heat kernel on R^d. We consider this problem for random and fractal media, giving conditions which allow a local limit theorem to be derived from weak convergence of a sequence of random walks on approximating graphs.
24/11/2009	Richard Melrose	MIT	Scattering and General Relativity
1/12/2009	Nicholas Burq		Dispersion for the water-wave equation This is a joint work with T. Alazard and C. Zuily (Orsay). In this work, we show how after a suitable paralinearization, the water-wave equation reduces to a system of Schrödinger type. This reduction allows then to prove dispersive estimates (smoothing or Strichartz type inequalities) by using rather standard semi-classical analysis
8/12/2009	Michael Eichmair	MIT/Monash

Seminars in Analysis and PDEs

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Title/Abstract