Browsing by Author "Mutavdžić, Nikola"
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Mutavdžić, Nikola (Beograd , 2023)[more][less]
Abstract: In this PhD thesis we investigate bounds of the gradient of harmonic and harmonic quasiconformal mappings. We also discuss such bounds for functions that are harmonic with respect to the hyperbolic metric or certain other metrics. This research has been motivated by some recent results about Lipschitz-continuity of quasiconformal mappings that satisfy the Laplace gradient inequality. More precisely, the mappings we consider are solutions of the Dirichlet problem for the Poisson equation and can be considered as a generalization of harmonic mappings. Besides the ball, we also work with general domains on which solutions of the Dirichlet problem are defined, as well as general codomains. Finally, we announce new results that have been formulated for regions of C1,α-smoothness, both as the domain and the codomain. Besides presenting the main results, we give an overview of general notions from differential geometry and recall some of the properties of hyperbolic metric in an n-dimensional ball. We also state properties of harmonic and sub-harmonic functions with respect to the hyperbolic metric, which are analogous to some classical results from the theory if harmonic functions and Hardy’s theory. It turns out that the gradients of hyperbolic harmonic functions behave differently from those of euclidean harmonic functions. A similar conclusion is obtained for the family of Tα-harmonic functions. Namely, unlike the space of harmonic functions, the solution of the Dirichlet problem in the space of Tα-harmonic functions is shown to be Lipschitz-continuous when so is the boundary function. In addition, we investigate Höldercontinuity of the solution of the Dirichlet problem for the Poisson equation in the euclidean and hyperbolic metric. We will present versions of the Schwarz lemma on the boundary for pluriharmonic mappings in Hilbert and Banach spaces. These results will follow from the version of the Schwarz lemma for harmonic mappings from the unit disc to the interval (1, 1) without the assumption that the point z = 0 maps to itself. Furthermore, we show a version of the boundary Schwarz lemma for harmonic mappings from a ball to a ball, not necessarily of the same dimension. The proof uses a version of the Schwarz lemma for multivariable functions, first considered by Burget. This result is obtained by integrating the Poisson kernel over so-called polar caps. The assumption that point z = 0 maps to itself is again not needed, thus yielding a generalization of a recent result by D. Kalaj. At the end of this section, it is demonstrated that the analogous result is false in the case of hyperbolic harmonic functions. In a certain sense, this means that the Hopf lemma is not valid for hyperbolic harmonic functions. Amongst various versions of the Schwarz lemma, we have been investigating bounds of the modulus for classes of holomorphic functions f on the unit disc whose index If fulfils certain geometric conditions. These classes are a generalization of the star and α-star functions, previously investigated by B. N. Örnek. Our method is based on using Jack’s lemma and can be applied in certain more general cases. As an illustration, we derive the sharp bounds for the modulus of a holomorphic function f with index If whose codomain is a vertical strip, as well as bounds for the modulus of the derivative of f at point z = 0. Moreover, we give a bound for the rate of growth of the modulus of holomorphic functions on disk U that map point z = 0 to itself and whose codomain is a vertical strip. URI: http://hdl.handle.net/123456789/5582 Files in this item: 1
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