Southern California
Analysis and Partial Differential Equations Conference

Titles and Abstracts

Titles and Abstracts

Mihai Putinar (UCSB) " Matrix models in
Laplacian Growth"

Abstract: A
specific
dynamics
of planar closed curves, roughly stated as velocity

equal
to
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

known
as
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.

Catherine
Sulem (Toronto) "Water waves over a rough bottom in
the shallow
water regime"

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
periodic
corrector term. Solutions of the latter can

exhibit the effect of Bragg resonance
with
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
the
scaling limit and the resulting error terms.

This is joint work with Walter Craig
and
David Lannes.

Lei Ni (UCSD) "Differential 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.

Betsy Stovall (UCLA) "Scattering for the cubic Klein--Gordon in 2 space dimensions"

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

Monica Visan.

Franc
Forstneric (Lubljana) "The Poletsky-Rosay theorem on singular
complex spaces"

Abstract: In the early 1990's Evgeny Poletsky
proved that the Poisson

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

Poletsky's
theorem to functions on any locally irreducible complex space X instead
of

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

modeling the evolution of small viscous droplets on a solid support (the

thin film equation). Along the contact line (aka the free boundary), the

solution must satisfy a gradient condition (contact angle condition).

While many existence and regularity results are known for solutions with

zero contact angle, the only existence result with non-zero contact angle is

due to Otto and only holds in some particular framework (Hele-Shaw cell).

Following Bertsch, Giacomelli and Karali, we consider a regularization of

this free boundary problem to attempt to generalize Otto's result.

Robert
Strain (Upenn) "Global solutions to a non-local diffusion
equation with quadratic non-linearity"

non-local diffusion equation. The initial condition is positive, radial, and non-increasing; these

conditions are propagated by the equation. There is however no size restriction on the initial data.

This model problem is of interest due to its structural similarity with Landau's equation from plasma

physics, and moreover its radically different behavior from the semi-linear Heat equation with

quadratic non-linearity. This is 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.