A minimal partition problem with trace constraint in the Grushin plane

TL;DR

This paper investigates a minimal perimeter partition problem with trace constraints in the Grushin plane, establishing existence and characterizing solutions, revealing differences from Euclidean geometry, and connecting to isoperimetric inequalities and minimal clusters.

Contribution

It introduces and analyzes a new variational problem in the Grushin plane, proving existence and characterizing solutions with trace constraints, highlighting differences from Euclidean cases.

Findings

Existence of regular solutions for the partition problem.

Characterization of solutions via isoperimetric sets.

Differences from Euclidean minimal partition problems.

Abstract

We study a variational problem for the perimeter associated with the Grushin plane, called minimal partition problem with trace constraint. This consists in studying how to enclose three prescribed areas in the Grushin plane, using the least amount of perimeter, under an additional "one-dimensional" constraint on the intersections of their boundaries. We prove existence of regular solutions for this problem, and we characterize them in terms of isoperimetric sets, showing differences with the Euclidean case. The problem arises from the study of quantitative isoperimetric inequalities and has connections with the theory of minimal clusters.