Related papers: More on Boundary Behavior of Univalent Harmonic Ma…
We study the behavior of the boundary function of a harmonic mapping from global and local points of view. Results related to the Koebe lemma are proved, as well as a generalization of a boundary behavior theorem by Bshouty, Lyzzaik and…
It is established that if a harmonic function $u$ on the unit disk $\mathbb D$ in $\mathbb C$ has angular limits on a measurable set $E$ of the unit circle $\partial\mathbb D$, then its conjugate harmonic function $v$ in $\mathbb D$ also…
We consider the boundary dynamics of iterated function systems of holomorphic self-maps of the unit disc. Our main result provides a sufficient condition which guarantees that the dynamical behaviour of a left iterated function system in…
We characterize the set of positive harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of several different chambers. We analyze the asymptotic behavior of the solutions in connection with the changes…
We construct sense-preserving univalent harmonic mappings which map the unit disk onto a domain which is convex in the horizontal direction, but with varying dilatation. Also, we obtain minimal surfaces associated with such harmonic…
This paper investigates the geometric and analytical properties of harmonic mappings $f$ in the unit disk $\mathbb{D}$ induced by boundary functions $F$ belonging to the Lebesgue spaces $L^{p}(\mathbb{T})$ for $1 \le p \le \infty$. We first…
Open discrete mappings with a modulus condition in metric spaces are considered. Some results related to local behavior of mappings as well as theorems about continuous extension to a boundary are proved.
The manuscript is devoted to the boundary behavior of mappings with bounded and finite distortion. We consider mappings of domains of the Euclidean space that satisfy weighted Poletsky inequality. Assume that, the definition domain is…
We study harmonic measure in finite graphs with an emphasis on expanders, that is, positive spectral gap. It is shown that if the spectral gap is positive then for all sets that are not too large the harmonic measure from a uniform starting…
We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with $0<\alpha<1$, in the vicinity of an exceptional boundary point where all such functions…
We obtain the sharp upper and lower bounds for the growth and distortion of the analytic parts $h$ of orientation-preserving harmonic mappings $f=h+\overline g$ (normalized in the standard way) that map the unit disk onto a convex domain.
The paper is devoted to the study of mappings with finite distortion, actively studied recently. For mappings whose inverse satisfy the Poletsky inequality, the results on boundary behavior in terms of prime ends are obtained. In…
We study the spectral behavior of higher order elliptic operators upon domain perturbation. We prove general spectral stability results for Dirichlet, Neumann and intermediate boundary conditions. Moreover, we consider the case of the…
The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled…
We analyze the boundaries of multiply connected Fatou components of transcendental maps by means of universal covering maps and associated inner functions. A unified approach is presented, which includes invariant Fatou components (of any…
Univalent functions are complex, analytic (holomorphic) and injective functions that have been widely discussed in complex analysis. It was recently proposed that the stringent constraints that univalence imposes on the growth of functions…
We develop a new technique for studying the boundary limiting behavior of a holomorphic function on a domain $\Omega$ -- both in one and several complex variables. The approach involves two new localized maximal functions. As a result of…
We study the boundary value problems for harmonic functions on open connected subsets of post-critically finite (p.c.f.) self-similar sets, on which the Laplacian is defined through a strongly recurrent self-similar local regular Dirichlet…
In this paper, we consider mappings on uniform domains with exponentially integrable distortion whose Jacobian determinants are integrable. We show that such mappings can be extended to the boundary and moreover these extensions are…
In this paper we prove that bounded Hua-harmonic functions on tube domains that satisfy some boundary regularity condition are necessarily pluriharmonic. In doing so, we show that a similar theorem is true on one-dimensional extensions of…