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

By A. Cialdea, Vladimir G. Maz'ya. 2004

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Abstract We prove that the algebraic condition | p − 2 | | 〈 Im A ξ , ξ 〉 | ⩽ 2 p − 1 〈 Re A ξ , ξ 〉 (for any ξ ∈ R n ) is necessary and sufficient for the L p -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 L p -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 L p -quasi-dissipativity of this operator. The same condition is necessary and sufficient for the L p -quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the L p -dissipativity in R n of the operator ∇ t ( A ∇ ) + b ∇ + a with constant coefficients.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/612b9369289a3123d4f3f626deb69c7860f72483