Riesz-type inequalities and overdetermined problems for triangles and quadrilaterals

TL;DR

This paper extends Riesz-type inequalities to polygons, showing that regular triangles and quadrilaterals maximize nonlocal energies among polygons with fixed area, and characterizes these polygons via overdetermined boundary problems.

Contribution

It establishes Riesz inequalities for polygons and characterizes regular polygons as solutions to overdetermined problems, extending classical results to discrete polygonal settings.

Findings

Regular triangles and quadrilaterals maximize nonlocal energies among polygons with fixed area.

Necessary conditions for polygons to be stationary points are derived.

Regular polygons are characterized as solutions to overdetermined boundary problems.

Abstract

We consider Riesz-type nonlocal interaction energies over polygons. We prove the analog of the Riesz inequality in this discrete setting for triangles and quadrilaterals, and obtain that among all $N$-gons with fixed area, the nonlocal energy is maximized by a regular polygon, for $N=3,4$. Further we derive necessary first-order stationarity conditions for a polygon with respect to a restricted class of variations, which will then be used to characterize regular $N$-gons, for $N=3,4$, as solutions to an overdetermined free boundary problem.