Steven R. Bell

Finitely generated function fields and complexity in potential theory in the plane

We prove that the Bergman kernel function associated to a finitely connected domain Ω in the plane is given as a rational combination of only three basic functions of one complex variable: an Alhfors map, its derivative, and one other function whose existence is deduced by means of the field of meromorphic functions on the double of Ω. Because many other functions of conformal mapping and potential theory can be expressed in terms of the Bergman kernel, our results shed light on the complexity of these objects. We also prove that the Bergman kernel is an algebraic function of a single Ahlfors map and its derivative. It follows that many objects of potential theory associated to a multiply connected domain are algebraic if and only if the domain is a finite branched cover of the unit disc via an algebraic holomorphic mapping.

Quadrature domains and kernel function zipping

It is proved that quadrature domains are ubiquitous in a very strong sense in the realm of smoothly bounded multiply connected domains in the plane. In fact, they are so dense that one might as well assume that any given smooth domain one is dealing with is a quadrature domain, and this allows access to a host of strong conditions on the classical kernel functions associated to the domain. Following this string of ideas leads to the discovery that the Bergman kernel can be “zipped” down to a strikingly small data set.It is also proved that the kernel functions associated to a quadrature domain must be algebraic.

On the classical Dirichlet problem in the plane with rational data

We consider the Dirichlet problem for the Laplace operator with rational data on the boundary of a planar domain. Our main results include a characterization of the disk as the only domain for which all solutions are rational and a characterization of the simply connected quadrature domains as the only ones for which all solutions are algebraic of a certain type.

Ahlfors maps, the double of a domain, and complexity in potential theory and conformal mapping

We prove that the Bergman kernel function associated to a finitely-connected planar domain can be expressed as a rational combination of two independent Ahlfors maps associated to the domain plus the derivative of one of the maps. Similar results are shown to hold for the Szegö and Poisson kernels and other objects of potential theory. These results generalize routinely to the case of a relatively compact domain in a Riemann surface with finitely many boundary components.

The Bergman Kernel and Quadrature Domains in the Plane

A streamlined proof that the Bergman kernel associated to a quadrature domain in the plane must be algebraic will be given. A byproduct of the proof will be that the Bergman kernel is a rational function of z and one other explicit function known as the Schwarz function. Simplified proofs of several other well known facts about quadrature domains will fall out along the way. Finally, Bergman representative coordinates will be defined that make subtle alterations to a domain to convert it to a quadrature domain. In such coordinates, biholomorphic mappings become algebraic.

Unique continuation theorems for the\(\bar \partial - Operator\) and applications

AbstractWe formulate a unique continuation principle for the inhomogeneous Cauchy-Riemann equations near a boundary pointz0 of a smooth domain in complex euclidean space. The principle implies that the Bergman projection of a function supported away fromz0 cannot vanish to infinite order atz0 unless it vanishes identically. We prove that the principle holds in planar domains and in domains where the

Density of quadrature domains in one and several complex variables

\bar \partial - Neumann

Algebraicity in the Dirichlet problem in the plane with rational data

problem is known to be analytic hypoelliptic. We also demonstrate the relevance of such questions to mapping problems in several complex variables. The last section of the paper deals with unique continuation properties of the Szegő projection and kernel in planar domains.