On extremal decomposition problems

Extremal decomposition problems in the Euclidean space

“…Following [2] we will say that a domainD is obtained by extending a domain D across a part of its boundary γ ⊂ ∂D if D ⊂D and (∂D) ∩D lies in γ. In [3] the notion of the Robin radius r (D, z 0 , Γ) = exp lim z→z 0 (g D (z, z 0 , Γ) + log |z − z 0 |) was introduced. This quantity generalized the notion of the conformal radius.…”

“…The second approach is developed mainly in works of V.N. Dubinin and his students [4,5,2,3]. In particular, a series of well-known results about extremal decomposition follows one way from composition principles for generalized reduced moduli (see [1, p. 56] and [12]).…”

“…on ∂D, where µ n−1 (∂D) is the area of boundary. By analogy with the definition of the Robin radius for plain domains from the paper [3] we will call the constant r(D, z 0 , Γ) the Robin radius of the domain D and the set Γ. Note that in the case of Γ = ∂D we get the harmonic radius [7,10,5].…”