The Beltrami Equation
A Geometric Approach
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Article
The Dirichlet problem for the Beltrami equations with sources
The paper is devoted to the study of the Dirichlet problem Re ω(z) → φ(ζ) as z → ζ, z ∈ D, ζ ∈ ∂D, with continuous boundary data φ : ∂D → ℝ for Beltrami equations
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Article
On the Hilbert problem for semi-linear Beltrami equations
The presented paper is devoted to the study of the well-known Hilbert boundary-value problem for semi-linear Beltrami equations with arbitrary boundary data that are measurable with respect to logarithmic capa...
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Article
BMO and Dirichlet Problem for Degenerate Beltrami Equation
Following Bojarski and Vekua, we have studied the Dirichlet problem lim ...
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Article
On the degenerate Beltrami equation and hydrodynamic normalization
The linear Beltrami equation on the Riemann sphere is studied under the assumption that its measurable complex-valued coefficient μ(z) has a compact support in ℂ and ‖μ‖∞ = 1 Sufficient conditions for the existen...
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Article
On the Hilbert boundary-value problem for Beltrami equations with singularities
We investigate the Hilbert boundary-value problem for Beltrami equations ∂ ¯ ...
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Article
Open AccessImpact of Comorbidities and Smoking on the Outcome in Aneurysmal Subarachnoid Hemorrhage
The intention of this observational study is to show the significant impact of comorbidities and smoking on the outcome in aneurysmal subarachnoid hemorrhage (SAH). During this observational study 203 cases of...
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Article
Open AccessChemotherapeutic xCT inhibitors sorafenib and erastin unraveled with the synaptic optogenetic function analysis tool
In the search for new potential chemotherapeutics, the compounds’ toxicity to healthy cells is an important factor. The brain with its functional units, the neurons, is especially endangered during the radio- ...
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Article
The Beltrami equations and prime ends
We first study the boundary behavior of ring Q-homeomorphisms in terms of Carathéodory’s prime ends and then give criteria to the solvability of the Dirichlet problem for the degenerate Beltrami equation ...
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Article
Selenium Action in Neuro-Oncology
The trace element selenium and selenocysteine-carrying selenoproteins play a pivotal role in the brain. Beside the essential function during development and maintenance of brain action, selenium has also been ...
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Article
Direct induction of haematoendothelial programs in human pluripotent stem cells by transcriptional regulators
Advancing pluripotent stem cell technologies for modelling haematopoietic stem cell development and blood therapies requires identifying key regulators of haematopoietic commitment from human pluripotent stem ...
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Book
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Chapter
The Degenerate Case
Recall that in our notation, a µ-homeomorphism in a domain D,D ⊂ ℂ is an ACL homeomorphic solution of (B) in D; see Sect. 1.5. For some functions µ with |µ(z)| ≤ 1 a.e. and ||µ||∞ = 1, there are no µ-homeomorphis...
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Chapter
On the Dirichlet Problem for Beltrami Equations
Boundary problems for the Beltrami equation (B) are due to the famous dissertation of Riemann who considered a particular case of analytic functions when (µz) ≡ 0 and to the works of Hilbert (1904, 1924) who stud...
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Chapter
Introduction
Let ℂ be the complex plane. In the complex notation w = u+iv and z = x+iy, the Beltrami equation in a domain D ⊂ ℂ has the form
B $$w_{... -
Chapter
The Classical Beltrami Equation ||µ||∞<1
Consider the Beltrami equation (B) in Sect. 1.1 with ||µ||∞<1, and let f : D → ℂ be its homeomorphic solution. Since |µ| < 1 a.e., f is sense preserving, and since ||µ||∞<1, f is a quasiconformal map**.
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Chapter
BMO- and FMO-Quasiconformal Map**s
In this chapter, the BMO-quasiconformal and BMO-quasiregular map**s in the plane are studied. This includes distortion, existence, uniqueness, representation, integrability, convergence and removability theo...
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Chapter
Strong Ring Solutions of Beltrami Equations
In this chapter we give a series of criteria for the existence of strong ring solutions, in particular, in terms of finite mean oscillation majorants for tangential dilatations. Moreover, we derive an extensio...
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Chapter
On the Beltrami Equations with Two Characteristics
The Beltrami equations of the first type
9.1.1 $$f_{\bar{z}} = \mu(z)f_{z}$$ is the basic equation for the theory of quas... -
Chapter
Preliminaries
The class BMO was introduced by John and Nirenberg in the paper [122] and soon became an important concept in harmonic analysis, partial differential equations, and related areas; see, e.g., [21, 24, 84, 103, ...
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Chapter
Ring Q-Homeomorphisms at Boundary Points
In this chapter, we introduce and study plane ring Q-homeomorphisms. This study is then applied to deriving general principles on the existence of strong ring solutions to the Beltrami equation extending and stre...
