Previous Special Year Seminar
On a Conjcture of J. Serrin
Haim Brezis
In 1964 J. Serrin proposed the following conjecture. Let u be a
weak solution (in W^{1,1}) of a second order elliptic equation in
divergence form, with Holder continuous coefficients, then u is a
"classical" solution ( i.e. u belongs to H^1). I will...
Curvature and Regularity of Optimal Transport
Curvature and Regularity of Optimal Transport
In 2005 Ma, Trudinger and Wang introduced a fourth-order
differential condition which comes close to be necessary and
sufficient for the smoothness of solutions to optimal transport
problems with a given cost function. If the cost function is
the...
$C^0$ Estimates for Conformally Invariant Equations on Locally Conformally Flat Manifolds with Umbilic Boundary
In recent years, fully nonlinear versions of the Yamabe problem
have received much attention. In particular, for manifolds with
boundary, $C^1$ and $C^2$a priori estimates have been proved for a
large class of data. To get an existence result, it is...
Asymptotic Curvature Decay of Bach-Flat Metrics
Jeff Streets
In pioneering work Tian/Viaclovsky initiated the study of the
moduli space of Bach-flat metrics. They showed C^0-orbifold
regularity and, equivalently, ALE order zero of noncompact
finite-energy solutions. By use of Kato inequalities, the
full...
Dispersion and Strichartz Type Estimates With No Loss for Schrodinger Equation in Trapping Geometries
I shall explain how to obtain Strichartz estimates with no loss
for Schrodinger equation in some cases where the geodesic flow has
some trapped trajectories, but the flow is hyperbolic. (This is
joint work with Burq and Hassell.)
Scalar Invariants for Even Dimensional Conformal Structures
The first aim of Fefferman-Graham ambient metric construction
was to write down all scalar invariants of conformal structures.
For odd dimensions, the aim was achieved with the aid of the
parabolic invariant theory by Bailey, Eastwood and Graham.
In...
Local Polyhomogeneity and Unique Continuation for Einstein Metrics
Olivier Biquard
I shall discuss two related local regularity results for
asymptotically hyperbolic (or complex hyperbolic) Einstein metrics,
near a point at infinity: local polyhomogeneity and unique
continuation.
Issues in Homogenization for Problems with Nondivergence Structure
Issues in Homogenization for Problems with Nondivergence Structure
When we look at a differential equation in a very irregular
media (composite material, mixed solutions, etc.) from very close,
we may see a very complicated problem. However, if we look from far
away we may not see the details and the problem may...