6 problems

Quadrature-domain critical-point conjecture. The number of critical points satisfies

Extremal critical-point conjecture. For every $n\geq 2$, and for each $k=1,2,\ldots,n+1$, such a domain exists. The bound is attained for $k=n+1$ by the extremal construction descr…

Let $n>2$. A quadrature domain in $\mathbb{R}^n$ is a domain satisfying a quadrature identity for the relevant class of functions, and an algebraic domain is one whose boundary is…

Let $\Omega$ be a subharmonic quadrature domain as above, with its associated function $u$ and boundary decomposition as in the preceding setup. Subharmonic quadrature-domain conje…

A quadrature domain is understood here in the classical, restricted sense, rather than in the wide sense allowing a compactly supported distribution in the quadrature formula. The…

Three-dimensional Neumann-oval conjecture. $W$ can be analytically continued throughout $\Omega\setminus B$. This is a proposed higher-dimensional analogue of the Schwarz potential…