Southern California Analysis and Partial Differential Equations Conference
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.)

Antoine Mellet (UMD) "A free boundary problem for thin films"

    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"

    Abstract: In this talk we will present our recent proof of the global in time well-posedness of a
                   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.