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Spectral minimal partitions of unbounded domains

Matthias Hofmann, James B. Kennedy, Hugo Tavares

TL;DR

This work extends spectral minimal partition theory to unbounded domains by replacing the first eigenvalue with the infimum λ(ω) of the spectrum of the Schrödinger operator −Δ+V/ω and introducing the essential-spectrum threshold Σ(Ω). It develops a relaxed, function-based variational framework and proves a sharp energy threshold 𝒯_{k,p}(Ω) that bounds the SMP energy, together with a concentration-compactness/IMS localization approach to obtain existence below the threshold. For p<∞, minimizers exist below threshold and are tied to ground states of the cells, while for p=∞ minimizers always exist (though not always as equipartitions at the threshold). The paper also proves regularity results for minimizers, continuity in p, and provides an extensive set of examples illustrating new phenomena unique to unbounded domains, thereby broadening the applicability of SMPs to more general geometric and spectral settings.

Abstract

We study the problem of constructing $k$-spectral minimal partitions of domains in $d$ dimensions, where the energy functional to be minimized is a $p$-norm ($1 \le p \le \infty$) of the infimum of the spectrum of a suitable Schrödinger operator $-Δ+V$, with Dirichlet conditions on the boundary of the partition elements (cells). The main novelty of this paper is that the domains may be unbounded, including of infinite volume. First, we prove a sharp upper bound for the infimal energy among all $k$-partitions by a threshold value which involves the infimum $Σ$ of the essential spectrum of the Schrödinger operator on the whole domain as well as the infimal energy among all $k-1$-partitions. Strictly below such threshold, we develop a concentration-compactness-type argument showing optimal partitions exist, and each cell admits ground states (i.e., the infimum of the spectrum on each cell is a simple isolated eigenvalue). Second, for $p<\infty$, when the energy and the threshold level coincide, we show there may or may not be minimizing partitions. Moreover, even when these exist, they may not have ground states. Third, for $p=\infty$, minimal partitions always exist, even at the threshold level, but these may or may not admit ground states. Moreover, below the threshold, we can always construct a minimizer, which is an equipartition. At the threshold value we show that spectral minimal partitions may not need to be equipartitions. We give a variety of examples of both domains and potentials to illustrate the new phenomena that occur in this setting.

Spectral minimal partitions of unbounded domains

TL;DR

This work extends spectral minimal partition theory to unbounded domains by replacing the first eigenvalue with the infimum λ(ω) of the spectrum of the Schrödinger operator −Δ+V/ω and introducing the essential-spectrum threshold Σ(Ω). It develops a relaxed, function-based variational framework and proves a sharp energy threshold 𝒯_{k,p}(Ω) that bounds the SMP energy, together with a concentration-compactness/IMS localization approach to obtain existence below the threshold. For p<∞, minimizers exist below threshold and are tied to ground states of the cells, while for p=∞ minimizers always exist (though not always as equipartitions at the threshold). The paper also proves regularity results for minimizers, continuity in p, and provides an extensive set of examples illustrating new phenomena unique to unbounded domains, thereby broadening the applicability of SMPs to more general geometric and spectral settings.

Abstract

We study the problem of constructing -spectral minimal partitions of domains in dimensions, where the energy functional to be minimized is a -norm () of the infimum of the spectrum of a suitable Schrödinger operator , with Dirichlet conditions on the boundary of the partition elements (cells). The main novelty of this paper is that the domains may be unbounded, including of infinite volume. First, we prove a sharp upper bound for the infimal energy among all -partitions by a threshold value which involves the infimum of the essential spectrum of the Schrödinger operator on the whole domain as well as the infimal energy among all -partitions. Strictly below such threshold, we develop a concentration-compactness-type argument showing optimal partitions exist, and each cell admits ground states (i.e., the infimum of the spectrum on each cell is a simple isolated eigenvalue). Second, for , when the energy and the threshold level coincide, we show there may or may not be minimizing partitions. Moreover, even when these exist, they may not have ground states. Third, for , minimal partitions always exist, even at the threshold level, but these may or may not admit ground states. Moreover, below the threshold, we can always construct a minimizer, which is an equipartition. At the threshold value we show that spectral minimal partitions may not need to be equipartitions. We give a variety of examples of both domains and potentials to illustrate the new phenomena that occur in this setting.

Paper Structure

This paper contains 14 sections, 15 theorems, 169 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Assumptions and notation
  3. Statement of main results
  4. Discussion and complementary results.
  5. On the assumptions on $\Omega$ and $V$
  6. On the threshold inequality \ref{['eq:optimal-inequality-p']}
  7. Monotonicity of the energy levels with respect to $p$ and $k$
  8. Description of examples
  9. Structure of the paper and strategy of the main proofs.
  10. Proof of Theorem \ref{['thm:firstmain']}
  11. Proof of Theorem \ref{['thm:secondmain']}: Existence
  12. Proof of Proposition \ref{['prop:p']}: Continuity in $p$
  13. Proof of Theorem \ref{['thm:regularity']}: Regularity
  14. Examples

Key Result

Theorem 1.3

For any $k\in \mathbb{N}$ and any $p\in [1,\infty)$, we have (where, by convention, for $k=0$ we set $\mathcal{L}_{0,p}(\Omega) := 0$). For $p=\infty$, there exists a (not necessarily connected) $k$-partition $\mathcal{P}$ such that

Figures (4)

  • Figure 6.1: A schematic representation of the construction of connected minimal partitions of the strip $(1,\infty) \times (0,\pi)$ for $k=2$ (left) and $k=3$ (right) (not to scale on the horizontal axis: the length of the "rooms" is chosen to increase towards infinity away from the left endpoint $x=1$). We introduce increasingly narrow passages to connect the rectangular regions (which will be of increasing length as $x$ increases). In order to treat the case $k\ge 4$ the domain $\Omega_2$ can be divided additionally (for $k=4$ indicated by the dotted line, note that of course the domains need to be rescaled).
  • Figure 6.2: An illustration of how for any given $k$, one can choose a $k$-partition consisting of $k-1$ wedges together with a central ball.
  • Figure 6.3: A representation of the domains in Example \ref{['ex:strip-ball']} and the corresponding minimizing three-partitions described there. These form equipartitions for which not all of the partition elements admit ground states.
  • Figure 6.4: The half-strip from Example \ref{['ex:fortschrittchen']}. The potential is taken to be $0$ to the left of $L$, and $c$ to its right.

Theorems & Definitions (45)

  • Remark 1.2
  • Theorem 1.3: Energy threshold
  • Theorem 1.4: Existence below the threshold
  • Remark 1.5
  • Theorem 1.6: Regularity of the minimizers
  • Proposition 1.7
  • Remark 1.8
  • Proposition 1.9: Equipartitions
  • Remark 1.10
  • Remark 1.11
  • ...and 35 more