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We prove the existence of a roof function for arclength null quadrature domains having finitely many boundary components. This bridges a gap toward classification of arclength null quadrature domains by removing an a priori assumption from previous classification results.
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Let $Omegasubsetmathbb{C}$ be a bounded domain. In this note, we use complex variable methods to study the number of critical points of the function $v=v_Omega$ that solves the elliptic problem $Delta v = -2$ in $Omega,$ with boundary values $v=0$ on $partialOmega.$ This problem has a classical flavor but is especially motivated by recent studies on localization of eigenfunctions. We provide an upper bound on the number of critical points of $v$ when $Omega$ belongs to a special class of domains in the plane, namely, domains for which the boundary $partialOmega$ is contained in ${z:|z|^2 = f(z) + overline{f(z)}},$ where $f(z)$ is a rational function. We furnish examples of domains where this bound is attained. We also prove a bound on the number of critical points in the case when $Omega$ is a quadrature domain, and conclude the note by stating some open problems and conjectures.
We describe recent work on the Bergman kernel of the (non-smooth) worm domain in several complex variables. An asymptotic expansion is obtained for the Bergman kernel. Mapping properties of the Bergman projection are studied. Irregularity properties of the kernal at the boundary are established. This is an expository paper, and considerable background is provided. Discussion of the smooth worm is also included.
This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.
We obtain uniform estimates for the canonical solution to $barpartial u=f$ on the Cartesian product of smoothly bounded planar domains, when $f$ is continuous up to the boundary. This generalizes Landuccis result for the bidisc toward higher dimensional product domains.
This is an auxiliary note to [12]. To be precise, here we have gathered the proofs of all the statements in [12, Section 5] that happen to have points of contact with techniques recently developed in Chousionis-Pratt [5] and Chunaev [6].
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