Riesztype inequalities and overdetermined problems for triangles and quadrilaterals
Abstract
We consider Riesztype 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 firstorder 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 arXiv:
 arXiv:2103.06657
 Bibcode:
 2021arXiv210306657B
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Optimization and Control;
 35N25;
 49Q10;
 49Q20;
 49J10;
 49J40;
 49K21
 EPrint:
 Convexity assumption removed