17th August 2009, 1510 MC206

**Dr. Gerd Schmalz**

from

UNE

will speak on

Solving the ∂-Problem Using Hölder Estimates in Non-Smooth Domains (20 Years Later)

The ∂-complex is the complex analogue of the de Rham complex in diﬀerential geometry. It gives information on the solvability of the ∂-problem: the p.d.e.

∂u=ƒ

for a diﬀerential form u with ƒ a given diﬀerential form on a domain in a complex manifold. In one complex dimension this reduces to the inhomogeneous Cauchy-Riemann equation

∂u/∂z=ƒ

In the 1950s Andreotti and Grauert proved that the solvability of the ∂-problem depends on convexity properties of the domain. Stronger versions of these results, with estimates on the solutions, were proved in the 1970s using integral formulæ for domains with smooth boundaries. My doctoral thesis generalised this to domains with non-smooth boundaries.