Minimizers of anisotropic perimeters with cylindrical norms

Giovanni Bellettini; Matteo Novaga; Shokhrukh Yusufovich Kholmatov

doi:10.3934/cpaa.2017068

Article Contents

2017, Volume 16, Issue 4: 1427-1454. Doi: 10.3934/cpaa.2017068

This issue Previous Article Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property Next Article Damping to prevent the blow-up of the korteweg-de vries equation

Minimizers of anisotropic perimeters with cylindrical norms

1.
Dipartimento di Ingegneria dell'Informazione e Scienze Matematiche, Università degli studi di Siena, via Roma 56,53100 Siena, Italy
2.
International Centre for Theoretical Physics (ICTP), Strada Costiera 11,34151 Trieste, Italy
3.
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5,56127 Pisa, Italy
4.
Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265,34136 Trieste, Italy

Received: April 2016

Revised: February 2017

Published: May 2017

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Abstract

We study various regularity properties of minimizers of the $\Phi$–perimeter, where $\Phi$ is a norm. Under suitable assumptions on $\Phi$ and on the dimension of the ambient space, we prove that the boundary of a cartesian minimizer is locally a Lipschitz graph out of a closed singular set of small Hausdorff dimension. Moreover, we show the following anisotropic Bernstein-type result: any entire cartesian minimizer is the subgraph of a monotone function depending only on one variable.

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Mathematics Subject Classification: Primary: 49Q05; Secondary: 53A10.

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Figure 1. $C_1^{(2)}(0, 0)\cup C_2^{(2)}(l, 0)$ with $l>0$ in Example 2.7(a) and its boundary. The right picture is a slight rotation of the left picture

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Figure 2. Union $C$ of $C_1^{(2)}(0, 0)$ and the $(-\pi/2)$-rotation of $C_2^{(2)}(0, 0)$ in Example 2.7(b). Notice that $C_t$ for $t=0$ is not a minimizer of the Euclidean perimeter in $\mathbb{R}^2$; however, this does not affect the minimality of $C$

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Figure 3. In case $0 < \gamma\leq l, $ among all sets connecting two components of $E$ the strip parallel to $\xi_1$-axis has the "smallest" $\Phi$-perimeter

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Figure 4. "Roof" like cone (left) and its section (right) along $(\partial H_1\cap\partial H_2)^\perp$

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Figure 5. Sections of cones when $\lambda_1 < +\infty$ and $\lambda_1=+\infty$

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