Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Periodic Delaunay cylinders with constant anisotropic nonlocal mean curvature

Francesc Alcover, Renzo Bruera

Abstract

In this article we prove existence and symmetry properties of periodic surfaces of revolution with constant anisotropic nonlocal mean curvature, generalizing a classical result of Delaunay to the anisotropic nonlocal setting. First, by studying the corresponding periodic isoperimetric problem, under natural assumptions on the kernel, we use rearrangement inequalities to extend a periodic version of the Wulff inequality to the nonlocal setting. This leads to the existence and symmetry properties of minimizers for every given volume in each period, thus generalizing the results of Cabré, Csató, and Mas to the anisotropic case. Second, under the same hypotheses on the kernel, we prove the existence of a one-parameter family of Delaunay near-cylinders in $\mathbb{R}^2$ bifurcating from a straight cylinder and having each constant anisotropic mean curvature. This extends the results of Cabré, Fall, Solà-Morales, and Weth to the anisotropic case. The stability of these near-cylinders will be studied in a forthcoming paper.

Periodic Delaunay cylinders with constant anisotropic nonlocal mean curvature

Abstract

In this article we prove existence and symmetry properties of periodic surfaces of revolution with constant anisotropic nonlocal mean curvature, generalizing a classical result of Delaunay to the anisotropic nonlocal setting. First, by studying the corresponding periodic isoperimetric problem, under natural assumptions on the kernel, we use rearrangement inequalities to extend a periodic version of the Wulff inequality to the nonlocal setting. This leads to the existence and symmetry properties of minimizers for every given volume in each period, thus generalizing the results of Cabré, Csató, and Mas to the anisotropic case. Second, under the same hypotheses on the kernel, we prove the existence of a one-parameter family of Delaunay near-cylinders in bifurcating from a straight cylinder and having each constant anisotropic mean curvature. This extends the results of Cabré, Fall, Solà-Morales, and Weth to the anisotropic case. The stability of these near-cylinders will be studied in a forthcoming paper.
Paper Structure (15 sections, 15 theorems, 205 equations, 1 figure)

This paper contains 15 sections, 15 theorems, 205 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Assumptions on the anisotropic kernel
  3. Kernels defined by a monotonic norm.
  4. Periodic minimizers with constant anisotropic nonlocal mean curvature via a variational approach
  5. Periodic near-cylinders with constant anisotropic nonlocal mean curvature in $\mathbb{R}^2$ through the implicit function theorem
  6. On the shape of minimizers.
  7. The periodic nonlocal perimeter.
  8. Existence and symmetry of constrained minimizers of the periodic anisotropic nonlocal perimeter
  9. Cylindrical and symmetric decreasing rearrangements
  10. Monotonicity under cylindrical and symmetric decreasing periodic rearrangements
  11. Existence of minimizers
  12. A family of curves in the plane with constant anisotropic nonlocal mean curvature through the implicit function theorem
  13. The functional setting
  14. The linearized operator acting on even periodic functions
  15. Differentiability of the nonlinear operator acting on even periodic functions

Key Result

Theorem 1.1

Let $n\geq 2$ and $K:\mathbb{R}^n \to (0,+\infty)$ be a positive measurable kernel satisfying H1::Symetry---H5::LowerBound, and let $\mathcal{P}_K$ be the associated periodic anisotropic nonlocal perimeter, as defined in eq:PANP_def. For every $\omega>0$, there exists a minimizer $E\subset \mathbb{R with $\Omega = \{x\in \mathbb{R}^n: -\pi < x_1 < \pi \}$. Moreover, up to sets of measure zero: As

Figures (1)

  • Figure 1: Pairs of interacting points in the definition of the anisotropic nonlocal perimeter of periodic (thick blue line) and non-periodic sets (thin orange line). The dotted vertical lines mark the boundary of the slab $\Omega=\{x\in \mathbb{R}^n : -\pi < x_1 <\pi\}$. The thick dashed blue and thin dashed orange lines correspond to possible competitors in the minimization of $\mathrm{Per}_K[\ \cdot \ ;\Omega]$ and $\mathcal{P}_K[\cdot]$, respectively.

Theorems & Definitions (27)

  • Theorem 1.1
  • Lemma 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • proof : Proof of Lemma \ref{['lemma_cylindrical']}
  • Theorem 2.3: BaernsteinTaylor1976, FriedbergLuttinger1976
  • proof : Proof of Theorem \ref{['theorem_monotonicity_symmetric_rearrangement']}
  • ...and 17 more