Criteria for the Lᵖ-dissipativity of systems of second order differential equations

الأصل · EN

We give complete algebraic characterizations of the Lᵖ-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form ∂ₕ(Ahk(x)∂ₖ), were Ahk(x) are m× m matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is Lᵖ-dissipative if and only if (1 2-1 p)² ≤ 2(ν-1)(2ν-1) (3-4ν)², ν being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the Lᵖ-dissipativity of the operator ∂ₕ (Aʰ(x)∂ₕ), where Aʰ(x) are m× m matrices with complex L¹ loc entries, and we describe the maximum angle of Lᵖ-dissipativity for this operator.