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).
File in questo prodotto:
File Dimensione Formato  
FMRS_isop_subfinsler.pdf

accesso aperto

Descrizione: Articolo
Tipologia: Preprint (submitted version)
Licenza: Accesso gratuito
Dimensione 2.21 MB
Formato Adobe PDF
2.21 MB Adobe PDF Visualizza/Apri
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11577/3460333
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 3
  • ???jsp.display-item.citation.isi??? 3
  • OpenAlex ND
social impact