Riesz-type inequalities and overdetermined problems for triangles and quadrilaterals

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.

This is a post-peer-review, pre-copyedit version of an article that will appear in the Journal of Geometric Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s12220-021-00737-7

References in corpus (2)

• Local and global minimality results for a nonlocal isoperimetric problem on R^N
• Minimality of polytopes in a nonlocal anisotropic isoperimetric problem

Cited by in corpus (2)

• Lattice tilings minimizing nonlocal perimeters
• Continuous symmetrizations and uniqueness of solutions to nonlocal equations