Research Output
1 - 10 of 17
Publication
Universal Taylor series, conformal mappings and boundary behaviour
2013-12, Gardiner, Stephen J.
A holomorphic function f on a simply connected domain Ω is said to possess a universal Taylor series about a point in Ω if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta K outside Ω (provided only that K has connected complement). This paper shows that this property is not conformally invariant, and, in the case where Ω is the unit disc, that such functions have extreme angular
boundary behaviour.
Publication
Two-phase quadrature domains
2012-01, Gardiner, Stephen J., Sjödin, Tomas
Recent work on two-phase free boundary problems has led to the investigation of a new type of quadrature domain for harmonic functions. This paper develops a method of constructing such quadrature domains based on the technique of partial balayage, which has proved to be a useful tool in the study of one-phase quadrature domains and Hele-Shaw flows.
Publication
Stationary Boundary Points for a Laplacian Growth Problem in Higher Dimensions
2014-08, Gardiner, Stephen J., Sjödin, Tomas
It is known that corners of interior angle less than π/2 in the boundary of a plane domain are initially stationary for Hele–Shaw flow arising from an arbitrary injection point inside the domain. This paper establishes the corresponding result for Laplacian growth of domains in higher dimensions. The problem is treated in terms of evolving families of quadrature domains for subharmonic functions.
Publication
Analytic content and the isoperimetric inequality in higher dimensions
2018-11-01, Gardiner, Stephen J., Ghergu, Marius, Sjödin, Tomas
This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in RN to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory.
Publication
Recent progress on fine differentiability and fine harmonicity
2012-12-15, Gardiner, Stephen J.
This paper describes recent results concerning the notions of differentiability and harmonicity with respect to the ne topology of classical potential theory.
Publication
Harmonic functions which vanish on a cylindrical surface
2016-01-15, Gardiner, Stephen J., Render, Hermann
Suppose that a harmonic function h on a finite cylinder vanishes on the curved part of the boundary. This paper answers a question of Khavinson by showing that h then has a harmonic continuation to the infinite strip bounded by the hyperplanes containing the flat parts of the boundary. The existence of this extension is established by an analysis of the convergence properties of a double series expansion of the Green function of an infinite cylinder beyond the domain itself.
Publication
Boundary Behaviour of Universal Taylor Series on Multiply Connected Domains
2014-10, Gardiner, Stephen J., Manolaki, Myrto
A holomorphic function on a planar domain Ω is said to possess a universal Taylor series about a point ζ of Ω if subsequences of the partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compact sets in C∖Ω that have connected complement. In the case where Ω is simply connected, such functions are known to be unbounded and to form a collection that is independent of the choice of ζ. This paper uses tools from potential theory to show that, even for domains Ω of arbitrary connectivity, such functions are unbounded whenever they exist. In the doubly connected case, a further analysis of boundary behaviour reveals that the collection of such functions can depend on the choice of ζ. This phenomenon was previously known only for domains that are at least triply connected. Related results are also established for universal Laurent series.
Publication
Boundary behaviour of Dirichlet series with applications to universal series
2016-10-05, Gardiner, Stephen J., Manolaki, Myrto
This paper establishes connections between the boundary behaviour of functions representable as absolutely convergent Dirichlet series in a half-plane and the convergence properties of partial sums of the Dirichlet series on the boundary. This yields insights into the boundary behaviour of Dirichlet series and Taylor series which have universal approximation properties.
Publication
Universal Taylor series for non-simply connected domains
2010-05, Gardiner, Stephen J., Tsirivas, Nikolaos
It is known that, for any simply connected proper subdomain Omega of the complex plane and any point zeta in Omega, there are holomorphic functions on Omega that have "universal" Taylor series expansions about zeta; that is, partial sums of the Taylor series approximate arbitrary polynomials on arbitrary compacta in C\Omega that have connected complement. This note shows that this phenomenon can break down for non-simply connected domains Omega, even when C\Omega is compact. This answers a question of Melas and disproves a conjecture of Müller, Vlachou and Yavrian.
Publication
Boundary behaviour of universal Taylor series
2014-02, Gardiner, Stephen J., Khavinson, Dmitry
A power series that converges on the unit disc D is called universal if its partial sums approx-
imate arbitrary polynomials on arbitrary compacta in CnD that have connected complement.
This paper shows that such series grow strongly and possess a Picard-type property near
each boundary point.