On the singular numbers of correct restrictions of non-selfadjoint elliptic differential operators

Abstract

Conditions are established on a correct restriction of an elliptic differential operator of order 2l defined on a bounded domain in ℝⁿ with sufficiently smooth boundary, ensuring that its singular numbers s_k are of order k^-2l/n. As an application certain estimates are obtained for the deviation upon domain perturbation of singular numbers of such correct restrictions. Let l, n ∈ ℕ and L be an elliptic differential expression of the following form: for u ∈ C∞(ℝⁿ),
(Lu)(x) = ∑_|α|,|β|≤l (-1)^|α|+|β|D^α (A_αβ(x)D^β u) , x ∈ ℝⁿ,
where A_αβ ∈ C^l(ℝⁿ) are real-valued functions for all multi-indices α, β satisfying |α|, |β| ≤ l. Moreover, let, for a domain Ω ⊆ ℝⁿ, L_Ω : D(L_Ω) → L₂(Ω) be a linear operator closed in L₂(Ω) generated by L on Ω.
A restriction A : D(A) → L₂(Ω) of L_Ω is correct if the equation Au = f has a unique solution u ∈ D(A) for any f ∈ L₂(Ω) and the corresponding inverse operator A^-1 : L₂(Ω) → D(A) is bounded. For the properties of correct restrictions see [5], [3].