Criterion for the Lp-dissipativity of second order differential operators with complex coefficients

AbstractWe prove that the algebraic condition |p−2||〈ImAξ,ξ〉|⩽2p−1〈ReAξ,ξ〉 (for any ξ∈Rn) is necessary and sufficient for the Lp-dissipativity of the Dirichlet problem for the differential operator ∇t(A∇), where A is a matrix whose entries are complex measures and whose imaginary part is symmetric. This result is new even for smooth coefficients, when it implies a criterion for the Lp-contractivity of the corresponding semigroup. We consider also the operator ∇t(A∇)+b∇+a, where the coefficients are smooth and ImA may be not symmetric. We show that the previous algebraic condition is necessary and sufficient for the Lp-quasi-dissipativity of this operator. The same condition is necessary and sufficient for the Lp-quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the Lp-dissipativity in Rn of the operator ∇t(A∇)+b∇+a with constant coefficients