What the study found
The study finds that for unbounded domains, including domains of infinite volume, spectral minimal partitions can exist below a threshold determined by the essential spectrum and the best “k-1” partition energy. The behavior at the threshold differs by the choice of p: for p < ∞, minimizing partitions may or may not exist, while for p = ∞ they always exist.
Why the authors say this matters
The authors present this as a way to extend spectral minimal partition theory to unbounded domains, which they say reveals new phenomena in this setting. The study suggests that the threshold behavior and existence of minimizers depend on the domain, the potential, and the choice of p.
What the researchers tested
The researchers studied k-spectral minimal partitions in d-dimensional domains, using a p-norm of the lowest value in the spectrum of a Schrödinger operator, -Δ + V, with Dirichlet boundary conditions on each partition cell. They proved a sharp upper bound, developed a concentration-compactness-type argument below the threshold, and gave examples of domains and potentials.
What worked and what didn't
Below the threshold, optimal partitions exist, and each cell has a ground state, meaning the lowest spectral value is a simple isolated eigenvalue. At the threshold, for p < ∞, minimizing partitions may exist or fail to exist, and even when they do exist, they may not have ground states; for p = ∞, minimal partitions always exist, but they may or may not have ground states. Below the threshold for p = ∞, the minimizer can always be chosen as an equipartition, while at the threshold minimal partitions need not be equipartitions.
What to keep in mind
The abstract does not give detailed assumptions on the specific domains or potentials beyond allowing unbounded domains and infinite volume. It also does not describe the full set of examples, so the range of cases is illustrated but not fully enumerated in the provided summary.
Key points
- The paper extends spectral minimal partition problems to unbounded domains, including infinite-volume domains.
- A sharp threshold is given in terms of the essential spectrum infimum and the best energy for k-1 partitions.
- Below the threshold, optimal partitions exist and each cell has a ground state.
- For p < ∞, threshold cases may or may not have minimizing partitions, and they may lack ground states.
- For p = ∞, minimal partitions always exist at the threshold, but they need not be equipartitions.
Disclosure
- Research title:
- Minimal partitions can exist in unbounded domains
- Authors:
- Matthias Hofmann, James B. Kennedy, Hugo Tavares
- Publication date:
- 2026-04-24
- OpenAlex record:
- View
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