3
$\begingroup$

Is ellipse a quadrature domain for arc-length measure? More precisely does there exist points $z_1,\cdots,z_n$ inside an ellipse $E$ and non zero constants $c_1,\cdots,c_n$ such that $$\int _Ep(z)\,ds_z=\sum_1^nc_kp(z_k)$$ holds for all polynomials $p$?

Motivation: In the literature, there are various results regarding characterization of domains in $\mathbb R^N$ via eigenvalues of integral (in particular potential) operators. A prime example is the following theorem of E. Fraenkel:

Theorem.(Fraenkel 2000) Let $G⊂\mathbb R^N$ be a bounded open set and let $ω_N$ be the surface measure of the unit-sphere in $\mathbb R^N$. Consider $$u(x)=\begin{cases} \frac{1}{2\pi}\int_G\log|x-y|dy, & n=2 \\ \frac{1}{N-2}\int_G\frac{1}{\|x-y\|^N-2}dy, & n\geq 3 \end{cases}.$$ If $u$ is constant on $∂G$ then $G$ is a ball.

One can ask similar question for the case that the operator on hand is defined by integration over boundary of closed curves against a singular kernel function, e.g. single layer potentials. Existence of quadrature formulas drastically simplifies the situation for such operators.

$\endgroup$
1
  • 1
    $\begingroup$ It might help MOers who want to think about this problem to know that there is significant literature about quadrature domains for arc-length, which I had never heard of before today. I'd recommend that anyone who wants to think about this problem do some reading to find out what is known. (E.g., start by doing a web search for papers on quadrature domains, which often mention literature on quadrature domains for arc-length.) $\endgroup$ Mar 20, 2015 at 1:34

1 Answer 1

1
$\begingroup$

No, the ellipse is not a quadrature domain for arclength. Using (see chapter 7 of Davis's book listed below) $dz = \frac{dz}{ds} ds$ and $\frac{dz}{ds} = \frac{1}{\sqrt{S'(z)}}$, where $S(z)$ denotes the Schwarz function of the ellipse (defined to be complex analytic and satisfying $S(z)=\bar{z}$ on the ellipse), we have $$\int_{E}g(z)ds = \int_{E} g(z) \sqrt{S'(z)} dz.$$ Then using, e.g., a modification of the proof of the second theorem in Chapter 14 of Davis's book listed below, if $E$ were a quadrature domain for arclength, then $\sqrt{S'(z)}$ would extend as a meromorphic function in the interior of the ellipse. But this is not the case as $S(z)$, and consequently $\sqrt{S'(z)}$ as well, has a branch cut singularity on the segment joining the foci of the ellipse.

Davis, Philip J., The Schwarz function and its applications, The Carus Mathematical Monographs. No. 17. Washington, D.C.: The Mathematical Association of America. XI, 228 p. (1974). ZBL0293.30001.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.