Let X and Y be planar Jordan domains of the same finite connectivity, Y being inner chordarc regular (such are Lipschitz domains). Every homeomorphism h:X->Y in the Sobolev space $W^{1,2}$ extends to a continuous map between closed domains. We prove that there exist homeomorphisms between closed domains which converge to h uniformly and in $W^{1,2}$. The problem of approximation of Sobolev homeomorphisms, raised by J. M. Ball and L. C. Evans, is deeply rooted in a study of energy-minimal deformations in nonlinear elasticity. The new feature of our main result is that approximation takes place also on the boundary, where the original map need not be a homeomorphism.
We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight $λ$ depends on the independent variable $z$. We prove that for an increasing radial weight $λ(z)$ the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight $λ(z)$ we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial s…
In this paper we first introduced a domain called generalized Dirichlet fundamental domain $\mathcal{F}^{*}$ for a Fuchsian group $G$ whose generators contain parabolic elements. This allows us to show that a quasisymmetric homeomorphism $h$ being compatible with a convergence Fuchsian group $G$ of first kind is a strongly quasisymmetric homeomorphism if and only if it has a quasiconformal extension $f$ to the upper half plane $\mathbb{H}$ onto itself such that the induced measure $λ_μ=|μ|^{2}/Im(z)dxdy$ by the Beltrami coefficient $μ$ of $f$ is a Carleson measure on the generalized Dirichlet …
We have proved theorems on compact classes of homeomorphisms with hydrodynamic normalization that are solutions of the Beltrami equation, whose characteristics are compactly supported and satisfy certain constraints of an integral type. As a consequence, we obtained results on compact classes of solutions of corresponding Dirichlet problems considered in some Jordan domain.
The present paper introduces the concept of monotone Hopf-harmonics in $2D$ as an alternative to harmonic homeomorphisms. It opens a new area of study in Geometric Function Theory (GFT). Much of the foregoing is motivated by the principle of non-interpenetration of matter in the mathematical theory of Nonlinear Elasticity (NE). The question we are concerned with is whether or not a Dirichlet energy-minimal mapping between Jordan domains with a prescribed boundary homeomorphism remains injective in the domain. The classical theorem of Radó-Kneser-Choquet asserts that this is the case when the t…
Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the classical Jordan-Schoenflies theorem may admit no solution - it is possible to have a boundary homeomorphism which admits a continuous $W^{1,2}$-extension but not even a homeomorphic $W^{1,1}$-extension. We prove that if the target is assumed to be a John disk, then any boundary homeomorphism from the unit circle admits a Sobolev homeomorphic extension for all exponents $p<2$. Jo…
We have proved theorems on compact classes of homeomorphisms with hydrodynamic normalization that are solutions of the Beltrami equation, whose characteristics are compactly supported and satisfy certain constraints of the theoretical-set type. As a consequence, we obtained results on compact classes of solutions of corresponding Dirichlet problems considered in some Jordan domain.
We consider conformal homeomorphisms $φ$ of generalized Jordan domains $U$ onto planar domains $Ω$ %, possibly {\bf infinitely connected}, that satisfy both of the next two conditions: (1) at most countably many boundary components of $Ω$ are non-degenerate and their diameters have a finite sum; (2) either the degenerate boundary components of $Ω$ or those of $U$ form a set of sigma-finite linear measure. We prove that $φ$ continuously extends to the closure of $U$ if and only if every boundary component of $Ω$ is locally connected. This generalizes the Carathéodory's Continuity Theorem and le…
We provide sufficient conditions so that a homeomorphism of the real line or of the circle admits an extension to a mapping of finite distortion in the upper half-plane or the disk, respectively. Moreover, we can ensure that the quasiconformal dilatation of the extension satisfies certain integrability conditions, such as $p$-integrability or exponential integrability. Mappings satisfying the latter integrability condition are also known as David homeomorphisms. Our extension operator is the same as the one used by Beurling and Ahlfors in their celebrated work. We prove an optimal bound for th…
