*Wissenschaftliches Programm*   *Liste der Vortragenden*

Sektion 2
Montag, 18.09.2000, 15.00–15.20 Uhr, POT 51

Lp-properties of second order elliptic differential operators

Hendrik Vogt, TU Dresden

The aim of the talk is to show how one can associate the (minus) generator of a positive C0-semigroup on Lp(_O_) with the formal differential expression

L =  - \~/  .(a \~/ ) + b1 . \~/  +  \~/  .b2 + V,

on an open set _O_  (_ RN N > 3, with singular measurable coefficients a: _O_ --> RN  ox RN ,  b 1, b2: _O_ --> RN ,  V : _O_ --> R. For simplicity, we restrict ourselves to the case that the matrix-valued function a is uniformly elliptic.

In the well-known case b1 = b2 = 0,  V > 0, one can associate an m-sectorial operator A2 in L2(_O_) 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 oo -contractive, so it extrapolates to a family of positive analytic semigroups T p(t) on Lp(_O_), for all p  (- [1,  oo ).

In the general case, the picture is quite different: On a certain interval I < [1,  oo ) obtained from the Lumer-Phillips theorem by a formal computation, L is associated with a quasi-contractive C0-semigroup T p(t) on Lp(_O_). 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).