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Correspondingly we also prove that any sequence of maximising domains $R_k^\\mathcal{N}$ for the Neumann eigenvalues $\\mu_k$ within the same collection of domains converges to the unit cube as $k\\to \\infty$. For $n=2$ this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1703.10249","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2017-03-29T21:39:20Z","cross_cats_sorted":[],"title_canon_sha256":"d9824aa671f49865414370ff2f9c324b438ea1c54facc1f94615ef0382085ae9","abstract_canon_sha256":"e650e93972fdc70ddd785b9580bb6b5c16f153cbaeb765bdeca9a8a758771a8f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:20.927432Z","signature_b64":"hfhiGDLa0N2Dmurdw2kjUhTPV8iTklH0WglTnTsEceYozrKoJwtofq3oC8YuPbFzAX7jSn59SK5m+gKQ7kZyCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5909cdbed08b1c3f9a029c6f32549ec8d9eb90024819802cba852734be0276c3","last_reissued_at":"2026-05-18T00:33:20.926933Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:20.926933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic behaviour of cuboids optimising Laplacian eigenvalues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Katie Gittins, Simon Larson","submitted_at":"2017-03-29T21:39:20Z","abstract_excerpt":"We prove that in dimension $n \\geq 2$, within the collection of unit measure cuboids in $\\mathbb{R}^n$ (i.e. domains of the form $\\prod_{i=1}^{n}(0, a_n)$), any sequence of minimising domains $R_k^\\mathcal{D}$ for the Dirichlet eigenvalues $\\lambda_k$ converges to the unit cube as $k \\to \\infty$. Correspondingly we also prove that any sequence of maximising domains $R_k^\\mathcal{N}$ for the Neumann eigenvalues $\\mu_k$ within the same collection of domains converges to the unit cube as $k\\to \\infty$. 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