Essential norms of the Neumann operator of the arithmetical mean

Title:

Essential norms of the Neumann operator of the arithmetical mean

Creator:

Král, Josef and Medková, Dagmar

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:dba1ab79-f00a-4848-bf38-dd675dc7297e
uuid:dba1ab79-f00a-4848-bf38-dd675dc7297e
doi:10.21136/MB.2001.134114

Subject:

double layer potential, Neumann’s operator of the arithmetical mean, and essential norm

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Let K ⊂ ℝm (m ≥ 2) be a compact set; assume that each ball centered on the boundary B of K meets K in a set of positive Lebesgue measure. Let C(1) 0 be the class of all continuously differentiable real-valued functions with compact support in m and denote by σm the area of the unit sphere in m. With each ϕ ∈ C(1) 0 we associate the function WKϕ(z) = 1⁄ σm ∫ Rm\K grad ϕ(x) · z − x |z − x| m dx of the variable z ∈ K (which is continuous in K and harmonic in K \ B). WKϕ depends only on the restriction ϕ|B of ϕ to the boundary B of K. This gives rise to a linear operator WK acting from the space C(1)(B) = {ϕ|B; ϕ ∈ C(1) 0 } to the space C(B) of all continuous functions on B. The operator TK sending each f ∈ C(1)(B) to TKf = 2WKf − f ∈ C(B) is called the Neumann operator of the arithmetical mean; it plays a significant role in connection with boundary value problems for harmonic functions. If p is a norm on C(B) ⊃ C(1)(B) inducing the topology of uniform convergence and G is the space of all compact linear operators acting on C(B), then the associated p-essential norm of TK is given by ωpTK = inf Q∈G sup {p[(TK − Q)f]; f ∈ C(1)(B), p(f) ≤ 1} . In the present paper estimates (from above and from below) of ωpTK are obtained resulting in precise evaluation of ωpTK in geometric terms connected only wit K.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

669-690

Source:

Mathematica bohemica | 2001 Volume:126 | Number:4

Harvested from:

CDK

Metadata only:

false