arXiv subjects

Tomas Sjödin

Publications and source records attributed to Tomas Sjödin.

7 recordsLinked to original sources

arXiv2026

Multiphase quadrature domains (existence and uniqueness)

The primary goal of this paper is to give a precise definition and prove existence and uniqueness of multiphase quadrature domains for subharmonic functions, ensuring that the prescribed measures are supported in the interior of the resulting domains. The approach to prove existence is based on a variational framework, where we minimize an energy functional over so called segregated states. In this respect we refine earlier results in this direction. But we also show that this approach alone is not enough for two reasons. First of all it seems hard to get existence results which ensure that the interior support condition is satisfied. And second it may happen, as we show by an example, that a multiphase quadrature domain exists but is not a minimizer of the energy functional. The main novelty of this work is the study of minimizers, if they exist, of the energy functional over a subset of the segregated states given by a natural constraint with respect to the given measures. From this approach we are able to prove uniqueness, and also give sufficient conditions for existence. We also give an example showing that, unlike the energy minimization and partial balayage approaches which are equivalent in the one-phase case, this equivalence breaks down already in the two-phase setting.

math.AP↗

arXiv2024

Partial balayage for the Helmholtz equation

Kow, Larson, Salo and Shahgholian recently initiated the study of quadrature domains for the Helmholtz equation and developed an associated theory of partial balayage of measures. The present paper offers an alternative approach to partial balayage in this context that yields stronger results. Applications are given to quadrature domains and to a domain evolution question that is analogous to Hele-Shaw flow.

math.AP↗

arXiv2019

Isoperimetric inequalities for Bergman analytic content

The Bergman $p$-analytic content ($1\leq p<\infty $) of a planar domain $Ω$ measures the $L^{p}(Ω)$-distance between $\overline{z}$ and the Bergman space $A^{p}(Ω)$ of holomorphic functions. It has a natural analogue in all dimensions which is formulated in terms of harmonic vector fields. This paper investigates isoperimetric inequalities for Bergman $p$-analytic content in terms of the St Venant functional for torsional rigidity, and addresses the cases of equality with the upper and lower bounds.

math.CA↗

arXiv2018

Analytic content and the isoperimetric inequality in higher dimensions

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

math.CA↗

arXiv2016

The Dirichlet problem for p-harmonic functions with respect to arbitrary compactifications

We study the Dirichlet problem for p-harmonic functions on metric spaces with respect to arbitrary compactifications. A particular focus is on the Perron method, and as a new approach to the invariance problem we introduce Sobolev-Perron solutions. We obtain various resolutivity and invariance results, and also show that most functions that have earlier been proved to be resolutive are in fact Sobolev-resolutive. We also introduce (Sobolev)-Wiener solutions and harmonizability in this nonlinear context, and study their connections to (Sobolev)-Perron solutions, partly using Q-compactifications.

math.AP↗

arXiv2016

A new approach to Sobolev spaces in metric measure spaces

Let $(X,d_X,μ)$ be a metric measure space where $X$ is locally compact and separable and $μ$ is a Borel regular measure such that $0 <μ(B(x,r)) <\infty$ for every ball $B(x,r)$ with center $x \in X$ and radius $r>0$. We define $\mathcal{X}$ to be the set of all positive, finite non-zero regular Borel measures with compact support in $X$ which are dominated by $μ$, and $\mathcal{M}=\mathcal{X} \cup \{0\}$. By introducing a kind of mass transport metric $d_{\mathcal{M}}$ on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for real valued functions $F$ on $\mathcal{X}$, and then for real valued functions $f$ on $X$ by identifying them with the unique function $F_f$ on $\mathcal{X}$ defined by the mean-value integral: $$F_f(η)= \frac{1}{\|η\|} \int f dη.$$ 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 $\mathbb{R}^n$ with Lebesgue measure.

math.AP↗

arXiv2011

Harmonic balls and two-phase Schwarz function

Here we shall introduce the concept of harmonic balls/spheres in sub-domains of $\R^n$, through a mean value property for a sub-class 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 can not have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exists, and gives rise to the generalization of the well-known Schwarz function to the case of 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.

math.AP↗