Series: Mathematics Colloquium

Date: Thursday, October 9, 2003

Time: 4:30 - 5:30 PM

Place: 102 McAllister Building

Host: Juan Gil (Altoona Campus), Victor Nistor

Refreshments: 4:00 - 4:30 PM, in 212 McAllister

Speaker: B.-W. Schulze, University of Potsdam

Title: 

Ellipticity in Pseudo-Differential Algebras on Manifolds with Corners

Abstract:  

Ellipticity of a differential operator on a closed smooth manifold is
equivalent to the Fredholm property between natural distribution
spaces.  In the first part of the lecture we give an introduction into
the basic ideas of operators associated with symbols (not necessarily
differential but also pseudo-differential ones, locally based on the
Fourier transform). We illustrate ellipticity in terms of principal
symbols and the pseudo-differential nature of parametrices. Moreover,
we show how this entails the Fredholm property. We see in this way
that the principal symbol of an elliptic operator which is in simplest
cases a scalar function (locally a polynomial in the case of a
differential operator) determines many interesting properties, e.g., a
finite index.  In the second part we ask the same things for the case
of boundary value problems, such as the Dirichlet or Neumann problem
for the Laplace operator in a smooth domain, and then also for
manifolds with a more complicated geometry, such as non-compact exits
or corners of different kind. We see that ellipticity in such cases is
often connected with a tuple of principal symbols, also containing
operator-valued components.  In smooth domains with boundary we have
two components, namely the former scalar (interior) symbol and a so
called boundary symbol. The ellipticity with respect to the second
component is also known as the Shapiro-Lopatinskij condition.
Finally, we briefly consider cases, where Shapiro-Lopatinskij elliptic
conditions are not possible for an operator with elliptic interior
symbol. The simplest example is the Cauchy-Riemann operator. Other
examples of that kind are Dirac operators. In such cases the
ellipticity near the boundary is organized in terms of global
projection conditions.  We also have a look at higher geometric
singularities which induce hierarchies of principal symbols and
combinations between higher versions of Shapiro-Lopatinskij
ellipticity and projection conditions.