Analysis and Partial Differential Equations Conference
Titles and Abstracts
Mihai Putinar (UCSB) " Matrix models in
of planar closed curves, roughly stated as velocity
harmonic measure, represents a mathematical idealization
of several natural physical phenomena.
Although studied for more than a
century, only ten years ago it was
proved that this moving interface is
governed by a completely integrable
system. Consequently, the influx of
ideas borrowed recently from modern
physics restructured the subject, previously
Laplacian Growth, of Hele-Shaw flows. A potential theoretic
discretization of this particular curve
evolution led to the asymptotic study
of the eigenvalue distribution of
random normal matrices, or of the finite
central sections of some close to
normal operators. The many mathematical
problems arising from this
discretization/approximation scheme will make the
main body of the lecture.
Sulem (Toronto) "Water waves over a rough bottom in
Abstract: I will present a study of the Euler
equations for free surface water waves in the case of
rapidly varying periodic bottom, in the
shallow water scaling regime. We derive a model
system of equations, consisting of the
classical shallow water equations coupled with
non-local evolution equations for a
corrector term. Solutions of the latter can
exhibit the effect of Bragg resonance
the periodic bottom, which leads to secular
growth and can influence the time
interval of validity of the theory. We justify the derivation
of the model with an analysis of
scaling limit and the resulting error terms.
This is joint work with Walter Craig
Lei Ni (UCSD)
Harnack estimates for differential forms"
Abstract: The Harnack estimates arises from
the celebrated De Gorgi-Nash-Moser
theory on the regularity of
solutions to the elliptic equations. In this talk I shall
explain a recent work on sharp
differential Harnack estimates for positive
forms on Kaehler manifolds.
Motivations from complex geometry and applications
shall be also discussed.
(UCLA) "Scattering for the cubic Klein--Gordon in 2 space
Abstract: We outline a recent proof that
finite energy, real-valued solutions to the defocusing equation cubic
Klein-Gordon equation in 2 space
dimensions scatter both forwards and backwards in time
and discuss a related result in
the focusing case. This is joint work with Rowan Killip and
Forstneric (Lubljana) "The Poletsky-Rosay theorem on singular
Abstract: In the early 1990's Evgeny Poletsky
proved that the Poisson
Antoine Mellet (UMD)
boundary problem for thin films"
envelope of an upper semicontinuous
function u on C^n is the largest
plurisubharmonic function v satisfying
v \le u. An immediate application is
the description of the polynomial hull
of a compact set by sequences of
analytic discs. In this joint work with
Barbara Drinovec Drnovsek we extend
theorem to functions on any locally irreducible complex space X instead
C^n. (The case when X is a complex
manifold was proved earlier by J.-P. Rosay.)
Abstract: The lubrication approximation leads to a
fourth order degenerate equation
viscous droplets on a solid support (the
contact line (aka the free boundary), the
condition (contact angle condition).
results are known for solutions with
existence result with non-zero contact angle is
holds in some particular framework (Hele-Shaw cell).
we consider a regularization of
attempt to generalize Otto's result.
Strain (Upenn) "Global solutions to a non-local diffusion
equation with quadratic non-linearity"
Abstract: In this talk we will present our recent proof
of the global in time well-posedness of a
initial condition is positive,
radial, and non-increasing; these
the equation. There is however no
size restriction on the initial data.
of interest due to its structural similarity with
Landau's equation from plasma
radically different behavior from the
semi-linear Heat equation with
a joint work with Joachim Krieger
Jeff McNeal (Ohio State)
"Extension of biholomorphic maps via non-holomorphic projections"
Abstract: I'll first show how a regularity property
on certain operators,
generalizing condition R of
Bell-Ligocka, implies smooth extension to the
boundary of biholomorphic mappings
between smoothly bounded, pseudoconvex
domains. I'll then show that this
regularity property always holds on this
class of domains, by showing that a
"twisted" version of the d-bar Neumann
problem is globally regular.