*Wissenschaftliches Programm* *Liste der Vortragenden*

**Sektion 2**

Montag, 18.09.2000,
15.00–15.20 Uhr, POT 51

#### Hendrik Vogt, TU Dresden

The aim of the talk is to show how one can associate the (minus) generator of a positive C_{0}-semigroup on
L_{p}() with the formal differential expression

on an open set ^{N }, N __>__ 3, with singular measurable coefficients a: ^{N } ^{N }, b_{
1}, b_{2}:
^{N }, V : . For simplicity, we restrict ourselves to the case that the matrix-valued function a is
uniformly elliptic.

In the well-known case b_{1} = b_{2} = 0, V __>__ 0, one can associate an m-sectorial operator A_{2} in L_{2}() with
L, by means of Kato’s first representation theorem for sectorial forms. Moreover, the positive analytic
semigroup T _{2}(t) := e^{-tA2} is L_{
1}- and L_{}-contractive, so it extrapolates to a family of positive analytic
semigroups T _{p}(t) on L_{p}(), for all p [1, ).

In the general case, the picture is quite different: On a certain interval I [1, ) obtained from the
Lumer-Phillips theorem by a formal computation, L is associated with a quasi-contractive C_{0}-semigroup
T _{p}(t) on L_{p}(). If a is uniformly elliptic, this interval can be extended to the left and to the right, but the
semigroup is no longer quasi-contractive outside I.

We present conditions on the lower order coefficients which imply that the semigroups T _{p}(t) are analytic
of p-independent angle, and the spectra of the corresponding generators are p-independent,
too.

This talk is based on a joint work with V. Liskevich and Z. Sobol (University of Bristol).