Localized boundary-domain singular integral equations of the Robin type problem for self-adjoint second-order strongly elliptic PDE systems

Definition A.1.

We say χ ∈ X k for an integer k ≥ 0 if

We say χ ∈ X + k for an integer k ≥ 1 if χ ∈ X k and σ χ ⁢ ( ω ) > 0 for all ω ∈ ℝ , where

We say χ ∈ X 1 + k for an integer k ≥ 1 if χ ∈ X + k and ω ⁢ χ ^ s ⁢ ( ω ) ≤ 1 for all ω ∈ ℝ .

Theorem B.1.

Let χ ∈ X k with k ≥ 3 have a compact support, and let t < k - 1 , and - 1 2 < s < k - 1 2 . Then the following localized operators are continuous:

Moreover, if χ ∈ X 1 + 3 , then the operator P : [ H r ⁢ ( R 3 ) ] 3 → [ H r + 2 ⁢ ( R 3 ) ] 3 is invertible for all r ∈ R and the inverse operator P - 1 is representable in the form

where Δ is the Laplace operator, I is the unit operator and M χ is a scalar pseudodifferential operator with the symbol m ^ χ ⁢ ( ξ ) satisfying the inequality

with some positive constant C.

Theorem B.2.

Let χ ∈ X 3 , ψ ∈ H - 1 / 2 ⁢ ( S ) , and φ ∈ H 1 / 2 ⁢ ( S ) . Then the following jump relations hold on S:

where

and d ⁢ ( y ) is positive definite due to (2.1).

Theorem B.3.

Let χ ∈ X k with k ≥ 3 and 1 < t < k - 1 . Then the following operators are continuous: