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On the anisotropic partitioning problem in Euclidean convex domains

César Rosales

TL;DR

The paper extends the anisotropic isoperimetric theory to Euclidean convex domains by developing a second-variation framework for the anisotropic area and proving the concavity of the anisotropic isoperimetric profile power $\psi=(I_{\Omega,K})^{(n+1)/n}$. It establishes existence and regularity of minimizers, examines the topology of minimizers including connectivity, and derives sharp comparisons with convex cones via $\mathcal{A}_K$-based second variation, yielding rigidity results when equality is achieved. It also analyzes anisotropic minimal hypersurfaces with free boundary, showing that stability and parabolicity constrain configurations to planar or conical geometries. Together, these results provide a comprehensive analytical and geometric framework for anisotropic isoperimetry in convex domains with potential applications to capillarity and material science.

Abstract

We consider the variational problem of minimizing an anisotropic perimeter functional under a volume constraint in a Euclidean convex domain. We extend to this setting analytical properties of the isoperimetric profile, topological features about the minimizers and sharp isoperimetric inequalities with respect to convex cones. Besides some geometric measure theory results about the existence and regularity of minimizers, the proofs rely on a second variation formula for the anisotropic area of hypersurfaces with non-empty boundary.

On the anisotropic partitioning problem in Euclidean convex domains

TL;DR

The paper extends the anisotropic isoperimetric theory to Euclidean convex domains by developing a second-variation framework for the anisotropic area and proving the concavity of the anisotropic isoperimetric profile power . It establishes existence and regularity of minimizers, examines the topology of minimizers including connectivity, and derives sharp comparisons with convex cones via -based second variation, yielding rigidity results when equality is achieved. It also analyzes anisotropic minimal hypersurfaces with free boundary, showing that stability and parabolicity constrain configurations to planar or conical geometries. Together, these results provide a comprehensive analytical and geometric framework for anisotropic isoperimetry in convex domains with potential applications to capillarity and material science.

Abstract

We consider the variational problem of minimizing an anisotropic perimeter functional under a volume constraint in a Euclidean convex domain. We extend to this setting analytical properties of the isoperimetric profile, topological features about the minimizers and sharp isoperimetric inequalities with respect to convex cones. Besides some geometric measure theory results about the existence and regularity of minimizers, the proofs rely on a second variation formula for the anisotropic area of hypersurfaces with non-empty boundary.

Paper Structure

This paper contains 9 sections, 24 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some facts about convexity
  4. Anisotropic extrinsic geometry
  5. Anisotropic area and perimeter
  6. The anisotropic partitioning problem
  7. Variational formulas and consequences
  8. Main results
  9. Anisotropic minimal hypersurfaces with free boundary

Key Result

Lemma 2.1

Let $\{C_i\}_{i\in\mathbb{N}}$ be a sequence of compact convex non-empty sets in $\mathbb{R}^{n+1}$ converging in Hausdorff distance to a compact convex non-empty set $C$. Then, we have

Theorems & Definitions (52)

  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • Proposition 3.6
  • ...and 42 more