MIT PDE/Analysis Seminar, Spring 2003
At MIT in Building 2
In Room 2-143 at 4:00 PM on Wednesdays, unless otherwise noted.
February 12: Eugene B. Dynkin (Cornell)
``Superdiffusions and positive solutions of semilinear partial differential
equations''
Abstract: A superdiffusion describes a random evolution of a cloud of
particles. It is closely related to equations involving an operator
$Lu-\psi(u)$ where L is a uniformly elliptic differential operator and
$\psi$ belongs to a class of functions which contains, in particular,
$\psi(u)=u^\alpha$ with $\alpha>1$. We are interested in describing all
positive solutions of the equation $Lu-\psi(u)=0$ in a smooth domain $D$ in
$R^d$. A special role is played by solutions which we call moderate: $u$
is moderate if there exists $h$ such that $u\le h$ and $Lh=0$. A solution
is called sigma-moderate if it is the limit of an increasing sequence of
moderate solutions.
In 1998 Dynkin and Kuznetsov introduced a concept of the fine boundary
trace of a solution $u$. They described all possible values of such
trace and they proved that a sigma-moderate solution is characterized
completely by its fine trace. A key problem remained open: is every
solution sigma-moderate? In 2002 Benoit Mselati (a PhD student of F. Le
Gall) gave a positive answer to this question in the case when $L$ is the
Laplacian and $\psi(u)=u^2$. Thus the problem of describing all positive
solutions of the equation $\Delta u=u^2$ in $D$ got a complete
solution. An analogous problem for more general equation still remains
open. It requires substantial new tools. Some progress in this direction
will be described in the talk.
February 19: Jeff Viaclovsky (MIT)
``A fully nonlinear equation on 4-manifolds with positive scalar
curvature''
Abstract: We present a conformal deformation involving a fully nonlinear
equation in dimension 4, starting with positive scalar curvature. Assuming
a certain conformal invariant is positive, one may deform from positive
scalar curvature to a stronger condition involving the Ricci tensor.
A special case of this deformation gives a more direct proof of the result
of Chang, Gursky and Yang. We also give a new conformally invariant
condition for positivity of the Paneitz operator, which allows us to
give many new examples of manifolds admitting metrics with constant
Q-curvature. This is joint work with Matt Gursky.
February 26: Mary Beth Ruskai (University of Massachusetts, Lowell)
``The Role of Entropy Inequalities in Quantum Information Theory''
Abstract:
In quantum computation and communication information is
encoded using vectors in C^{2^n} rather than strings
of 0 and 1. Many questions arise about the extent to
which this richer structure can be used to enhance the
encoding, transmission and extraction of information.
The von Neumann entropy plays an important role in this
analysis.
This talk will discuss a group of entropy inequalities,
the role they play in quantum information theory, and
aspects of their proofs. If time permits, some connections
to operator algebras and p-norms will be discussed.
March 5: Mohammad Ghomi (University of South Carolina)
``Locally convex hypersurfaces and Monge-Ampere equations''
Abstract:
We will give a survey of some recent results concerning the geometry
and topology of positively curved hypersurfaces with boundary in
Euclidean space. These include an optimal regularity theorem for
the boundary of convex hulls of submanifolds, the speaker's joint
work with S. Alexander on a convex hull property for positively
curved surfaces which is dual to that of the negatively curved case,
and a convergence theorem for locally convex surfaces which has been
used recently by Guan and Spruck in their work on existence of
surfaces with constant positive curvature and prescribed boundary.
March 12: Vadim Kaloshin (AIM, IAS, Caltech)
``On Newhouse phenomenon (of infinitely many coexisting sinks)''
Abstract:
Consider the space of $C^r$ diffeomorphisms (smooth invertible selfmaps)
of a compact surface $M$ (e.g. $S^2$ or $T^2$) Diff$^r(M)$ with $r\geq 2$.
A sink of $f:M \to M$ is a periodic point $x \in M$ which attract all
points from its neighbourhood (as in your kitchen). Points attracted to
$x$ called basin of attraction of $x$. In 60-th Thom conjectured
that a generic diffeomorphism can not have infinitely many coexisting
sinks. Indeed, each sink has an open basin of attraction and infinitely
many of those seems too much. In 70-th Newhouse constructed an open set
of diffeomophisms $N \subset \textup{Diff}^r(M)$ such that generic
diffeomorphism in $N$ does have infinitely many coexisting sinks.
It is an amazing phenomenon, called Newhouse phenomenon. It disproves
Thom's conjecture and significant obstacle to describe ergodic properties
of surface diffeomorphisms. We shall discuss this phenomenon and
closely related results of Benedicks-Carleson, Mora-Viana, Wang-Young,
Morreira-Yoccoz. The main result indicates in some sense this phenomenon
has "probability zero". This is a particular case of so-called
Palis conjecture.
March 19: Neshan Wickramasekera (MIT)
``A rigidity theorem for stable, minimal hypercones''
Abstract:
Consider the weak (i.e. varifold) limit of a sequence of smooth, stable
minimal hypersurfaces of dimension n immersed in Euclidean space.
Suppose the limit varifold is a cone. By Allard's regularity theorem,
one has that if the vertex density of the cone is sufficiently close to 1,
then the cone must be a hyperplane. Now consider the case when
the density of the cone is greater than or equal to 2 and close to 2.
In this talk, I will present a proof that if such a cone is also
sufficiently close to a pair of hyperplanes
(two transverse hyperplanes or a single hyperplane with multiplicity 2)
then it must be equal to a pair of hyperplanes. I will also discuss some
applications of this theorem to low density singularities of a general
varifold that is the weak limit of a sequence of immersed, stable
hypersurfaces.
April 2 at 3:30pm in 2-143: Sergiu Klainerman (Princeton)
``On the causal structure of rough
Einstein metrics''
April 9 at 3:30pm in 2-143: Felix Otto (Universität Bonn)
``Structure of entropy solutions''
Abstract: Entropy solutions of conservation laws are a priori just bounded
and measurable, but one thinks of them as being piecewise smooth. We show
that in case of a scalar but multidimensional conservation law, the
solutions are indeed piecewise smooth in the sense that the discontinuity
set is rectifiable. The crucial assumption on the equation is genuine
nonlinearity, a simple geometric condition on the characteristics. The
proof relies on the geometric classification of blow--ups in the framework
of the kinetic formulation (Perthame, Lions, Tadmor). This is joint work
with C. De Lellis and M. Westdickenberg.
April 16: Two talks:
3:30pm: Moritz Kassmann (University of Conneticut)
``Harmonic Functions for Nonlocal Operators of Variable Order''
Abstract: We investigate regularity properties of functions that are harmonic
with respect to nonlocal operators of variable order. These
non-scale-invariant operators arise as generators of Markovian jump
processes. Our primary aim is to extend the Harnack inequality by Krylov and
Safonov to the operators under consideration. Although the oscillation over
balls turns out to be dependent on the radius we are able to prove the
Harnack inequality and the continuity
of harmonic functions. The talk is based on ongoing joint research together
with R. Bass.
4:30pm: Vladimir Maz'ya (Linköping University and Northeastern University)
``Wiener's test for higher order elliptic equations''
Abstract:
Wiener's test for the regularity of a boundary point with
respect to the Dirichlet problem for the Laplace equation
is extended to ellliptic partial differential equations of an
arbitrary even order. Some open problems are discussed.
April 23: Dimitry Jakobson (McGill University)
``Some results on L^p norms and
critical points of eigenfunctions''
Abstract:
We construct a metric of revolution on the 2-torus and a sequence
of eigenfunctions with growing eigenvalues and exactly 16 critical points.
We also discuss relations between positive and negative parts of
eigenfunctions on Riemannian manifolds.
April 30:
May 7: Hart Smith (University of Washington)
``L^p estimates for quasimodes in bounded planar domains''
Abstract:
I will discuss joint work with Sogge obtaining best possible L^p
estimates for quasimodes (functions with spectrum localised to
a unit frequency interval) on 2 dimensional manifolds with boundary.
The estimates that hold on boundary-free manifolds are known to
break down near points where the boundary is convex. Our work
combines techniques developed to handle quasimode estimates on
manifolds with Lipschitz metrics, together with control of the
reflected geodesic flow.
May 14: Irina Mitrea (Cornell University)
``Mellin Transform and Global Optimization Techniques for Elliptic
Boundary Problems''
Abstract:
We employ Mellin transform techniques, Calderón-Zygmund
theory and global optimization techniques in the study of
spectral properties of layer potential operators associated
to elliptic systems in two dimensions. Our analysis includes
the case of the Lamé system of elastostatics, the Stokes
system of hydrostatics and second order elliptic systems in
in canonical form.