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

الأصل · EN

We prove that the algebraic condition |p-2| |< Im Aξ,ξ>| ≤ 2 √p-1 < Re Aξ,ξ> (for any ξⁿ) 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 Im A 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.