We study the isoperimetric problem for anisotropic perimeter measures on R3, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm ϕ on the horizontal distribution. In the case where ϕ is the standard norm in the plane, such isoperimetric problem is the subject of Pansu’s conjecture, which is still unsolved. Assuming some regularity on ϕ and on its dual norm ϕ∗, we characterize C 2-smooth isoperimetric sets as the sub-Finsler analogue of Pansu’s bubbles. The argument is based on a fine study of the characteristic set of ϕ-isoperimetric sets and on establishing a foliation property by sub-Finsler geodesics. When ϕ is a crystalline norm, we show the existence of a partial foliation for constant ϕ-curvature surfaces by sub-Finsler geodesics. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where ϕ is crystalline).

The Isoperimetric Problem for Regular and Crystalline Norms in H1

Franceschi V.
;
Monti R.;Righini A.;
2023

Abstract

We study the isoperimetric problem for anisotropic perimeter measures on R3, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm ϕ on the horizontal distribution. In the case where ϕ is the standard norm in the plane, such isoperimetric problem is the subject of Pansu’s conjecture, which is still unsolved. Assuming some regularity on ϕ and on its dual norm ϕ∗, we characterize C 2-smooth isoperimetric sets as the sub-Finsler analogue of Pansu’s bubbles. The argument is based on a fine study of the characteristic set of ϕ-isoperimetric sets and on establishing a foliation property by sub-Finsler geodesics. When ϕ is a crystalline norm, we show the existence of a partial foliation for constant ϕ-curvature surfaces by sub-Finsler geodesics. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where ϕ is crystalline).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3460333
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