The isoperimetric problem for regular and crystalline norms in ${\mathbb{H}}^1$

Abstract

We study the isoperimetric problem for anisotropic left-invariant perimeter measures on ${\mathbb{R}}^3$, endowed with the Heisenberg group structure. The perimeter is associated with a left-invariant norm $\varphi$ on the horizontal distribution. We first prove a representation formula for the $\varphi$-perimeter of regular sets and, assuming some regularity on $\varphi$ and on its dual norm ${\varphi }^{\ast }$, we deduce a foliation property by sub-Finsler geodesics of ${\mathrm{C}}^2$-smooth surfaces with constant $\varphi$-curvature. We then prove that the characteristic set of ${\mathrm{C}}^2$-smooth surfaces that are locally extremal for the isoperimetric problem is made of isolated points and horizontal curves satisfying a suitable differential equation. Based on such a characterization, we characterize ${\mathrm{C}}^2$-smooth $\varphi$-isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. We also show, under suitable regularity properties on $\varphi$, that such sub-Finsler candidate isoperimetric sets are indeed ${\mathrm{C}}^2$-smooth. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where $\varphi$ is crystalline).

References in corpus (2)

• Periodic controls in step 2 sub-Finsler problems
• Pansu-Wulff shapes in ${\mathbb{H}}^1$