Tomas Sjödin

Convexity and the Exterior Inverse Problem of Potential Theory

Let Ω 1 and Ω 2 be bounded solid domains such that their associated volume potentials agree outside Ω 1 U Ω 2 . Under the assumption that one of the domains is convex, it is deduced that Ω 1 =Ω 2 .

Mother bodies of algebraic domains in the complex plane

We give a definition of a mother body of a domain in the complex plane, and prove some continuity properties of its potential in terms of the Schwarz function (which is explicitly assumed to exist). We end the article by studying the case of the ellipse, and use the previous results to prove existence and uniqueness of a mother body in this case, as well as a related existence result about graviequivalent measures for the ellipse.

Quadrature Identities and Deformation of Quadrature Domains

We study the possibility of deforming quadrature domains into each other, and also discuss the possibility of changing the distribution in a quadrature identity from complex to real and from real to positive. The last question is in a sense also studied without the assumption that we have a quadrature domain.

Harmonic balls and the two-phase Schwarz function

In this article, we introduce the concept of harmonic balls in sub-domains of ℝ n , through a mean-value property for a subclass of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions cannot have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exist, and gives rise to the generalization of the well-known Schwarz function to the case of a two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.

Quadrature Domains and Their Two-Phase Counterparts

This survey describes recent advances on quadrature domains that were made in the context of the ESF Network on Harmonic and Complex Analysis and its Applications (2007–2012). These results concern quadrature domains, and their two-phase counterparts, for harmonic, subharmonic and analytic functions.

Potential Theory in Denjoy Domains

This paper presents an account of Denjoy domains in relation to minimal harmonic functions, the boundary behaviour of the Green function, and to their usefulness as a source of counterexamples in potential theory. The discussion begins with an exposition of key work of Ancona and Benedicks and then moves on to describe several very recent results.

Analytic content and the isoperimetric inequality in higher dimensions

Abstract This paper establishes a conjecture of Gustafsson and Khavinson, which relates the analytic content of a smoothly bounded domain in R N to the classical isoperimetric inequality. The proof is based on a novel combination of partial balayage with optimal transport theory.

A new approach to Sobolev spaces in metric measure spaces

Let (X, dX , μ) be a metric measure space where X is locally compact and separable and μ is a Borel regular measure such that 0 < μ(B(x, r)) < ∞ for every ball B(x, r) with center x ∈ X and radius r > 0. We define X to be the set of all positive, finite non-zero regular Borel measures with compact support in X which are dominated by μ, and M = X ∪ {0}. By introducing a kind of mass transport metric dM on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions F : X → R, and then for functions f : X → [−∞,∞] by identifying them with the unique element Ff : X → R defined by the mean-value integral: Ff (η) = 1 ‖η‖ ∫ f dη. In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space R with Lebesgue measure.