Schwarz lemma for harmonic mappings between Riemann surfaces
We prove a Schwarz type lemma for harmonic mappings between the unit and a geodesic line in a Riemenn surface.
math.CV↗
arXiv subjects
David Kalaj
Publications and source records attributed to David Kalaj.
Neohookean deformations of annuli in the higher dimensional Euclidean space
Let $n\ge 2$ be an integer and assume that $\mathbb{A}=\{x\in\mathbf{R}^n:1<|x|<R\}$ and $\A_\ast = \{y \in \mathbf{R}^n: 1 < |y| < R_\ast\}$ be two annuli in Euclidean space $\mathbf{R}^n$. Assume that $\mathcal{F}(\A, \A_\ast)$ (resp. $\mathcal{R}(\A, \A_\ast)$) be the class of all orientation preserving (resp. radial) homeomorphisms $h : \A \mapsto \A_\ast$ in the Sobolev space $\mathcal{W} ^{1,n}(\A, \A_\ast)$ which keep the boundary circles in the same order. In this paper, we extended the corresponding results of Iwaniec and Onninen which was published in {\it Math. Ann.} Vol. 348, 2010.
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Lipschitz property of minimisers between double connected surfaces
We study the global Lipschitz character of minimisers of the Dirichlet energy of diffeomorphisms between doubly connected domains with smooth boundaries from Riemann surfaces. The key point of the proof is the fact that minimisers are certain Noether harmonic maps, with Hopf differential of special form, a fact invented by Iwaniec, Koh, Kovalev and Onninen in \cite{inv} for Euclidean metric and by the author in \cite{calculus} for the arbitrary metric, which depends deeply on a result of Jost \cite{Job1}.
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On Heinz type inequality for the half-plane and Gaussian curvature of Minimal surfaces
We prove a Heinz type inequality for harmonic diffeomorphisms of of the half-plane onto itself. We then apply this result to prove some sharp bound of the Gaussian curvature of a minimal surface, provided that it lies above the whole half-plane in $\mathbf{R}^3$.
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Harmonic maps between two concentric annuli in $\mathbf{R}^3$
Given two annuli $\mathbf{A}(r,R)$ and $\mathbf{A}(r_\ast, R_\ast)$, in $\mathbf{R}^3$ equipped with the Euclidean metric and the weighted metric $|y|^{-2}$ respectively, we minimize the Dirichlet integral, i.e. the functional $\mathscr{F}[f] = \int_{\mathbf{A}(r,R)} \frac{\Vert Df\Vert^2} {|f|^2}$, where $f$ is a homeomorphism between $\mathbf{A}(r,R)$ and $\mathbf{A}(r_\ast,R_\ast)$, which belongs to the Sobolev class $\mathscr{W}^{1,2}$. The minimizer is a certain generalized radial mapping, i.e. a mapping of the form $f(|x|η)=ρ(|x|)T(η)$, where $T$ is a conformal mapping of the unit sphere onto itself. It should be noticed that in this case no Nitsche phenomenon occur.
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Hyperelastic deformations and total combined energy of mappings between annuli
We consider the so called combined energy of a deformation between two concentric annuli and minimize it, provided that it keep order of the boundaries. It is an extension of the corresponding result of Euclidean energy. It is intrigue that, the minimizers are certain radial mappings and they exists if and only if the annulus on the image domain is not too thin, provided that the original annulus is fixed. This in turn implies a Nitsche type phenomenon. Next we consider the combined distortion and obtain certain related results which are dual to the results for combined energy, which also involve some Nitche type phenomenon. {The main part of the paper is concerned with the total combined energy, a certain integral operator, defined as a convex linear combination of the combined energy and combined distortion, of diffeomorphisms between two concentric annuli $\A(1,r)$ and $\B(1,R)$. First we construct radial minimizers of total combined energy, then we prove that those radial minimizers are absolute minimizers on the class of all mappings between the annuli under certain constraint. This extends the main result obtained by Iwaniec and Onninen in \cite{arma}.}
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Approximation of Kolmogorov-Smirnov Test Statistics
Motivated by the weak limit of the Kolmogorov-Smirnov test statistics, in this contribution, we concern the asymptotics of \begin{align*} \mathbb{P}\left\{\sup_{\boldsymbol{x}\in [0,1]^n}\left(W(\boldsymbol{x})\Big| W(\boldsymbol{1})=w\right)>u\right\}, \ w\in\mathbb{R}, \end{align*} for large $u$ where $W(\boldsymbol{x})$ is the multivariate Brownian sheet based on a distribution function $F$. The results related to general $F$ are investigated and some important examples are also showed.
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Quasiconformal mappings and Hölder continuity
We establish that every $K$-quasiconformal mapping $w$ of the unit ball $\IB$ onto a $C^2$-Jordan domain $Ω$ is Hölder continuous with constant $α= 2-\frac{n}{p}$, provided that its weak Laplacean $Δw$ is in $ L^p(\IB)$ for some $n/2<p<n$. In particular it is Hölder continuous for every $0<α<1$ provided that $Δw\in L^n(\IB)$.
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The Schwarz type Lemmas and the Landau type theorem of mappings satisfying Poisson's equations
For a given continuous function $g:~Ω\rightarrow\mathbb{C}$, we establish some Schwarz type Lemmas for mappings $f$ in $Ω$ satisfying the {\rm PDE}: $Δf=g$, where $Ω$ is a subset of the complex plane $\mathbb{C}$. Then we apply these results to obtain a Landau type theorem, which is a partial answer to the open problem in \cite{CP}. %The obtained results are
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A sharp inequality for harmonic diffeomorphisms of the unit disk
We extend the classical Schwarz-Pick inequality to the class of harmonic mappings between the unit disk and a Jordan domain with given perimeter. It is intriguing that the extremals in this case are certain harmonic diffeomorphisms between the unit disk and a convex domain that solve the Beltrami equation of second order.
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$(n,ρ)-$harmonic mappings and energy minimal deformations between annuli
We extend the main results obtained by Iwaniec and Onninen in Memoirs of the AMS (2012). In the paper it is solved the minimization problem of $(ρ,n)$ energy of Sobolev homeomorphisms between two concentric annuli in the Euclidean space $\mathbf{R}^n$. Here $ρ$ is a radial metric defined in the image annulus. The key of the proofs comes from the solution to the Euler-Lagrange equation for radial harmonic mapping. This is a new contribution on the topic of famous Nitsche conjecture.
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Lipschitz continuity of holomorphic mappings with respect to Bergman metric
n this paper, we establish the sharp estimate of the Lipschitz continuity with respect to the Bergman metric. The obtained results are the improvement and generalization of the corresponding results of Ghatage, Yan and Zheng (Proc. Amer. Math. Soc., 129: 2039-2044, 2000).
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Total energy of radial mappings
The main aim of this paper is to extend one of the main results of Iwaniec and Onninen (Arch. Ration. Mech. Anal., 194: 927-986, 2009). We prove that, the so called total energy functional defined on the class of radial streachings between annuli attains its minimum on a total energy diffeomorphism between annuli. This involves a subtle analysis of some special ODE.
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Gradient of solution of the Poisson equation in the unit ball and related operators
In this paper we determine the $L^1\to L^1$ and $L^{\infty}\to L^\infty$ norms of an integral operator $\mathcal{N}$ related to the gradient of the solution of Poisson equation in the unit ball with vanishing boundary data in sense of distributions.
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On Riesz type inequalities for harmonic mappings on the unit disk
We prove some sharp inequalities for complex harmonic functions on the unit disk. The results extend a M. Riesz conjugate function theorem and some well-known estimates for holomorphic functions. We apply some of results to the isoperimetric inequality for harmonic mappings.
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On some Riesz and Carleman type inequalities for harmonic functions on the unit disk
We prove some isoperimetric type inequalities for real harmonic functions in the unit disk belonging to the Hardy space $h^p$, $p>1$ and for complex harmonic functions in $h^4$. The results extend some recent results on the area. Further we discus some Riesz type results for holomorphic functions.
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On holomorphic functions with cluster sets of finite linear measure
We prove that if f is a holomorphic function on the open unit disc in C whose cluster set C(f) has finite linear measure and is such that the complement of C(f) has finitely many components, then the derivative of f belongs to the Hardy space H^1.
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Superharmonicity of Logarithm of Jacobian of harmonic mappings between surfaces
We prove that the Logarithm of the Jacobian of a sense preserving harmonic mappings between surfaces is superharmonic, provided that the Gaussian curvature of the image domain is non-negative.
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