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The $\ell_1$ double-bubble problem in three dimensions

Manuel Friedrich, Wojciech Górny, Ulisse Stefanelli

Abstract

We characterize the unique minimizer of the three-dimensional double-bubble problem with respect to the $\ell_1$-norm for volume ratios between $1/2$ and $2$.

The $\ell_1$ double-bubble problem in three dimensions

Abstract

We characterize the unique minimizer of the three-dimensional double-bubble problem with respect to the -norm for volume ratios between and .
Paper Structure (10 sections, 7 theorems, 128 equations, 2 figures)

This paper contains 10 sections, 7 theorems, 128 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of the main result
  3. The planar case
  4. Notation and optimal lower bound
  5. Statements of auxiliary results
  6. Proof of Theorem \ref{['thm:mainresult']}
  7. Proofs of the auxiliary results
  8. Proof of Lemma \ref{['lem:symmetrization-new']}
  9. Proof of Lemma \ref{['lem:psmaller13']}
  10. Proof of Lemma \ref{['lem:appendix']}

Key Result

Theorem 1.1

Letting ${V_B}/{V_A} \in[1/2,2]$, the unique minimizer of the double-bubble problem eq:db_problem are two cuboids sharing a square face. Up to translation and axis-preserving isometries, the minimizer can be specified as The minimal energy is given by

Figures (2)

  • Figure 1: The unique minimizer of the double-bubble problem \ref{['eq:db_problem']}.
  • Figure 2: Minimizers for the planar double-bubble problem \ref{['eq:2D_energy']}.

Theorems & Definitions (10)

  • Theorem 1.1: Characterization of the minimizer
  • Proposition 2.1: Characterization of the planar minimizer
  • Proposition 2.2: Minimal planar energy
  • Proposition 2.3: Optimal lower bound
  • Lemma 2.4: Slicing lemma
  • Lemma 2.5: Upper bound on $p$
  • Lemma 2.6: Functions $g_A^r$ and $g_B^r$
  • proof : Proof of Proposition \ref{['prop:estimateforenergy']}
  • proof : Proof of Theorem \ref{['thm:mainresult']}
  • proof : Proof of Lemma \ref{['lem:symmetrization-new']}