Related papers: Harmonic quasiconformal mappings between $\mathscr…

A K-quasiconformal selfmap of the unit disk with identity boundary values satisfies the H\"older estimate $|f(z)-f(w)| \leq 4^{1-1/K} |z-w|^{1/K}$. The constant $4^{1-\frac{1}{K}}$ is sharp.

We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain $\Omega \subset \mathbb{R}^2$ and prove higher integrability of the gradient up to the boundary by incorporating…

Conformal mapping may be the best-known topic in complex analysis. Any simply connected nonempty domain $\Omega$ in the complex plane ${{\mathbb{C}}}$ (assuming $\Omega\ne {{\mathbb{C}}}$) can be mapped bijectively to the unit disk by an…

For a fixed constant $\lambda > 0$ and a bounded Lipschitz domain $\Omega \subset \mathbb{R}^n$ with $n \geq 2$, we establish that almost-minimizers (functions satisfying a sort of variational inequality) of the Alt-Caffarelli type…

In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates):…

In this paper, we prove that the identity map for the smoothly bounded pseudoconvex domain of finite type in $\mathbb{C}^2$ extends to a bi-H\"{o}lder map between the Euclidean boundary and Gromov boundary. As an application, we show the…

We give a complete classification of local and global conformal biharmonic maps between any two space forms by proving that a conformal map between two space forms is proper biharmonic if and only if the dimension is 4, the domain is flat,…

We study mappings satisfying some estimate of distortion of modulus of families of paths. Under some conditions on definition and mapped domains, we have proved that these mappings are logarithmic H\"{o}lder continuous at boundary points.

We prove that proper pseudo-holomorphic maps between strictly pseudoconvex regions in almost complex manifolds extend to the boundary. The key point is that the Jacobian is far from zero near the boundary, and the proof is mainly based on…

In this article, we consider the H\"{o}lder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the H\"{o}lder continuity of the indicated class…

In this paper we study quasiconformal curves which are a special case of quasiregular curves. Namely embeddings $\Omega\rightarrow\mathbb{R}^m$ from some domain $\Omega\subset\mathbb{R}^n$ to $\mathbb{R}^m$, where $n\leq m$, which belong in…

In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain a sharp growth theorem of the radial length of…

In this article, we first discuss the Lipschitz characteristic and the linear measure distortion of $K$-quasiconformal harmonic mappings. Then we give some characterizations of the radial John disks with the help of Pre-Schwarzian of…

In this paper we present a geometric proof of the following fact. Let $D$ be a Jordan domain in $\mathbb{C}$, and let $f$ be analytic on $cl(D)$. Then there is an injective analytic map $\phi:D\to\mathbb{C}$, and a polynomial $p$, such that…

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 extend \emph{Schramm's cofat uniformization theorem} to cofat domains on upper Ahlfors 2-regular metric two-spheres $X$. Specifically, we show that if $\Omega \subset X$ is a cofat domain, then there exists a…

We show that for uniform domains $\Omega\subseteq \mathbb{R}^{d+1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to $\alpha$-dimensional Hausdorff…

We prove that two proper holomorphic polynomial maps between bounded symmetric domains of classical type which preserve the origin are equivalent if and only if they are isotropically equivalent.

We consider Jordan curves of the form $\gamma=\cup_{j=1}^n \gamma_j$ on the Riemann sphere for which each $\gamma_j$ is a hyperbolic geodesic in $(\widehat{\mathbb C} \smallsetminus \gamma)\cup \gamma_j$. These Jordan curves are…

All the known counterexamples to Kac' famous question "can one hear the shape of a drum", i.e., does isospectrality of two Laplacians on domains imply that the domains are congruent, consist of pairs of domains composed of copies of…