Mathematical problems of dynamical interaction of fluids and multiferroic solids

Classes of Kernels and Continuity Properties of the Double Layer Potential in Hölder Spaces

“…We now consider the seventh addendum in the right hand side of formula (4.10). Since B 1 is analytic, Lemma 4.5 (ii) and the product Proposition 3.6 (iii) imply that DB 1 (x − y)a (2) ν(y) belongs to K 0,1,1…”

“…Then the product Theorem 3.5 (ii) and Proposition 3.6 (iii) imply that the kernel (2) ν(y) belongs to the class K n−1,n,1 (∂Ω × ∂Ω). By the imbedding Proposition 3.7 (ii) with…”

“…Then one can prove the following formula for the kernel of the double layer potential B * Ω,y (S a (x − y)) = −DS a (x − y)a (2) ν(y) − ν t (y)a (1) S a (x − y) In particular, the kernel B * Ω,y (S a (x − y)) belongs to K ,(∂Ω)×(∂Ω) for all ∈]0, +∞[.…”

“…, n} is the canonical basis of R n . We note that the matrix a (2) is symmetric. Then we assume that a ∈ C N2 satisfies the following ellipticity assumption inf…”

“…In case n ≥ 2, m ≥ 2, α ∈]0, 1], O. Chkadua [2] has pointed out that one could exploit Kupradze, Gegelia, Basheleishvili and Burchuladze [22, Chap. IV, Sect.…”