Equality cases for condenser capacity inequalities under symmetrization

Abstract

Abstract. We prove that any first order F 2 Mm 1 ,m 2 ,n 1 ,n 2 -natural operator transforming projectable general connections on an (m1, m2, n1, n2The aim of this paper is to describe all F 2 M m 1 ,m 2 ,n 1 ,n 2 -natural operators transforming projectable general connections on an (m 1 , m 2 , n 1 , n 2 )-dimensional fibred-fibred manifolds into general connections on the vertical prolongation V Y → M of p : Y → M . The similar problem for the case of fibred manifolds was solving in [7]. In the paper [1], authors described natural operators transforming connections on fibred manifoldsA fibred-fibred manifold is a fibred surjective submersion

Infinitesimally small spheres and conformally invariant metrics

“…Since every end point x ∈ S(x * , r) of the radial segment from x * to x, which is in the spherical cone Φ, satisfies condition (3.7) it follows that µ D (x 0 , x) is constant on Φ. Obviously, Φ has interior points and the latter conclusion contradicts the fact established in part (1) of this proof that µ(x) can not have relative minimum in B(x 0 , R 0 ) \ {x 0 }. Thus, our assumption was wrong and µ D (x 0 , x) does not have relative maxima in B(x 0 , R 0 ).…”

“…In dimension n = 2, the cases when equality occurs in (3.2) were discussed under a variety of assumptions in [4], [13], and [1]. Also, in dimensions n ≥ 3, the cases of equality in polarization inequalities for the Newtonian capacity were discussed in [13] and [1]. In dimensions n ≥ 3, the question on the cases when equality sign occurs in (3.2), i.e.…”

Equality case for an elliptic area condenser inequality and a related Schwarz type lemma

“…W. Gehring gives a proof of the above in [7], also showing that the same is true if we consider the hyperbolic or the elliptic areas instead of the Euclidean. The equality conditions were studied in [3] for the Euclidean and in [2] for the hyperbolic case. We will prove an analogous condition for the elliptic case.…”

Πυκνωτές Και Γεωμετρικοί Μετασχηματισμοί

“…Ïé áðïäåßîåéò ôùí ÈåùñçìÜôùí 6, 7 êáé 8 äßíïíôáé óôï ÊåöÜëáéï 6 (âë. åðßóçò [14]). ç ïðïßá ÷áñáêôçñßaeåé ôéò K-çìéêáíïíéêÝò (quasiregular) áðåéêïíßóåéò óôï åðßðåäï (âë.…”