On the $L^p$-theory of $C_0$-semigroups associated with second-order elliptic operators with complex singular coefficients

Abstract

We study $L^p$-theory of second-order elliptic divergence type operators with complex measurable coefficients. The major aspect is that we allow complex coefficients in the main part of the operator, too. We investigate generation of analytic $C_0$-semigroups under very general conditions on the coefficients, related to the notion of form-boundedness. We determine an interval $J$ in the $L^p$-scale, not necessarily containing $p=2$, in which one obtains a consistent family of quasi-contractive semigroups. This interval is close to optimal, as shown by several examples. In the case of uniform ellipticity we construct a family of semigroups in an extended range of $L^p$-spaces, and we prove $p$-independence of the analyticity sector and of the spectrum of the generators.