Tomas Sjödin: Two-phase quadrature domains and harmonic balls

This talk is mainly going to be a survey of the recent theory of two-phase quadrature domains and the related topic of harmonic balls.
In particular I will focus on my work together with Stephen Gardiner (UCD Dublin) regarding two-phase quadrature domains for harmonic and subharmonic functions and my work with Henrik Shahgholian (KTH) about two-phase quadrature domains for analytic functions and harmonic balls.

Roughly speaking, two-phase quadrature domains consists of a pair of disjoint open sets $D_+,D_-$ together with two measures $\mu_+,\mu_-$ such that $\mu_+$ and $\mu_-$ has compact support in $D_+$ and $D_-$ respectively, and such that we for some suitable class of functions $h$ have an integral equality or inequality between the integrals (where $\lambda$ denotes Lebesgue measure) $$\int_{D_+}h d \lambda - \int_{D_-} h d \lambda,$$ and $$\int h d \mu_+ - \int h d \mu_-.$$ Natural choices for $h$ can be analytic, harmonic or such that $h$ is subharmonic in $D_+$ and superharmonic in $D_-$. Unlike the classical (one-phase) case we assume more about the behaviour of the functions at the boundaries of $D_+$ and $D_-$, and not just that $h$ is integrable over $D_+$ and $D_-$ (otherwise we would just have two disjoint one-phase quadrature domains). We will discuss what natural choices are, and also relate this concept to two-phase modified Schwarz potentials and Schwarz functions which also has natural definitions.

After this we shall also discuss the concept of harmonic balls, which is closely related to the above. It is well known that if $\alpha \delta_x$ is a point mass at $x \in \R^n$ and $B$ is an open set such that the Newtonian potential of $\alpha \delta_x$ and $\lambda |_B$ are equal in the complement of $B$, then $B$ is the ball with center $x$ and total mass $\alpha$. Harmonic balls are defined relative to a domain $K$, and we say that $B \subset K$ is a harmonic ball with respect to $\alpha \delta_x$ ($x \in K$) if the Green potentials in $K$ for $\alpha \delta_x$ and $\lambda |_B$ agree in $K \setminus B$. We will discuss some known results and also some interesting open questions regarding these.