Abstract
We give a complete characterization of all isoperimetric sets contained in a domain of the Euclidean plane, that is bounded by a Jordan curve and satisfies a no neck property. Further, we prove that the isoperimetric profile of such domain is convex above the volume of the largest ball contained in it, and that its square is globally convex.
Highlights
Given a bounded, open set ⊂ Rn, n ≥ 2, we consider the isoperimetric problem among Borel subsets of, that is, the minimization of the perimeter P(E) of a Borel set E ⊂ subject to a volume constraint |E| = V, where by perimeter we mean the distributional one in the sense of Caccioppoli–De Giorgi, and where |E| denotes the Lebesgue measure of E and V ∈ [0, | |]
If R denotes the inradius of and if ωn represents the Lebesgue measure of the unit ball in Rn, the classical isoperimetric inequality in Rn implies that the unique minimizers for volumes 0 < V ≤ ωn Rn are balls, up to null sets
Finding and characterizing minimizers, as well as computing J (V ), is a trivial problem whenever V ≤ ωn Rn, while it becomes a challenging problem for larger V
Summary
Open set ⊂ Rn, n ≥ 2, we consider the isoperimetric problem among Borel subsets of , that is, the minimization of the perimeter P(E) of a Borel set E ⊂ subject to a volume constraint |E| = V , where by perimeter we mean the distributional one in the sense of Caccioppoli–De Giorgi, and where |E| denotes the Lebesgue measure of E and V ∈ [0, | |]. We are interested in the properties of the total isoperimetric profile. If R denotes the inradius of (i.e., the radius of the largest ball contained in ) and if ωn represents the Lebesgue measure of the unit ball in Rn, the classical isoperimetric inequality in Rn implies that the unique minimizers for volumes 0 < V ≤ ωn Rn are balls, up to null sets.
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