The aim of the talk is to show how one can associate the (minus) generator of a positive C0-semigroup on Lp() with the formal differential expression
on an open set N , N > 3, with singular measurable coefficients a: N N , b 1, b2: N , V : . 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() 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 Lp(), 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 C0-semigroup T p(t) on Lp(). 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).