In this talk, I will discuss some characterizations of Sobolev
spaces, BV spaces, and present some new inequalities in this
context. As a consequence, I can improve classical properties of
Sobolev spaces such as Sobolev inequality, Poincare...
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...
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...
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...
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...
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.)
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...
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.
