Calculus of Variations and Geometric Measure Theory

M. Bonacini - R. Cristoferi - I. Topaloglu

Riesz-type inequalities and overdetermined problems for triangles and quadrilaterals

created by topaloglu1 on 11 Mar 2021
modified on 02 Jul 2022


Published Paper

Inserted: 11 mar 2021
Last Updated: 2 jul 2022

Journal: J. Geom. Anal.
Volume: 32
Year: 2022

ArXiv: 2103.06657 PDF


We consider Riesz-type nonlocal interaction energies over convex 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.

Keywords: shape optimization, Riesz's rearrangement inequality, polygons, overdetermined problem, Polya and Szego conjecture