Quasi-symmetric mappings and their generalizations on Riemannian manifolds

A relation between $\eta$-quasi-symmetric homomorphisms and $K$-quasiconformal mappings on $n$-dimensional smooth connected Riemannian manifolds has been studied. The main results of the research are presented in Theorems 2.6 and 2.7. Several conditions for the boundary behavior of $\eta$-quasi-symmetric homomorphisms between two arbitrary domains with weakly flat boundaries and compact closures, QED and uniform domains on the Riemannian mani\-folds, which satisfy the obtained results, were also formulated. In addition, quasiballs, $c$-locally connected domains, and the corresponding results were also considered.

On a connection between some classes of mapping on Riemannian manifolds

In this paper we study the connection between $\eta$-quasisymmetric homomorphisms and $K$-quasi\-con\-for\-mal mappings on $n$-dimensional smooth connected Riemannian manifolds. The main result of our research is the Theorem 3.1. For its proof we use a partition of unity method, which subordinate to the locally finite atlas of the manifold. Several results on the boundary behavior of $\eta$-quasisymmetric homomorphisms between two arbitrary domains, QED (uniform) domains and domains with weakly flat boundaries and compact closures on the Riemannian manifolds are also obtained in view of the above relations. The obtained results can be applied to Finsler manifolds with the addition of some conditions, which will take into account the specific of the initial manifold.

Compactness of 𝑀-uniform domains and optimal thermal insulation problems

In this paper, we will consider an optimal shape problem of heat insulation introduced by [D. Bucur, G. Buttazzo and C. Nitsch, Two optimization problems in thermal insulation, Notices Amer. Math. Soc. 64 (2017), 8, 830–835]. We will establish the existence of optimal shapes in the class of 𝑀-uniform domains. We will also show that balls are stable solutions of the optimal heat insulation problem.

Approximation by uniform domains in doubling quasiconvex metric spaces

We show that any bounded domain in a doubling quasiconvex metric space can be approximated from inside and outside by uniform domains.