The standard lens cluster in ℝ² uniquely minimizes relative perimeter

In this article we consider the isoperimetric problem for partitioning the plane into three disjoint domains, one having unit area and the remaining two having infinite area. We show that the only solution, up to rigid motions of the plane, is a lens cluster consisting of circular arcs containing the finite area region, attached to a single axis, with two triple junctions where the arcs meet at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="120 Superscript ring"> <mml:semantics> <mml:msup> <mml:mn>120</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∘</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">120^{\circ }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> angles. In particular, we show that such a configuration is a local minimizer of the total perimeter functional, and on the other hand any local minimizer of perimeter among clusters with the given area constraints must coincide with a lens cluster having this geometry. Some known results and conjectures on similar problems with both finite and infinite area constraints are presented at the conclusion.