Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues

Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues We prove that in dimension $$n \ge 2$$ n ≥ 2 , within the collection of unit-measure cuboids in $$\mathbb {R}^n$$ R n (i.e. domains of the form $$\prod _{i=1}^{n}(0, a_n)$$ ∏ i = 1 n ( 0 , a n ) ), any sequence of minimising domains $$R_k^\mathcal {D}$$ R k D for the Dirichlet eigenvalues $$\lambda _k$$ λ k converges to the unit cube as $$k \rightarrow \infty $$ k → ∞ . Correspondingly we also prove that any sequence of maximising domains $$R_k^\mathcal {N}$$ R k N for the Neumann eigenvalues $$\mu _k$$ μ k within the same collection of domains converges to the unit cube as $$k\rightarrow \infty $$ k → ∞ . For $$n=2$$ n = 2 this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for $$n=3$$ n = 3 was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as $$k \rightarrow \infty $$ k → ∞ . We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first k eigenvalues. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

Asymptotic Behaviour of Cuboids Optimising Laplacian Eigenvalues

Loading next page...
 
/lp/springer_journal/asymptotic-behaviour-of-cuboids-optimising-laplacian-eigenvalues-QptiWpt0JP
Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-017-2407-5
Publisher site
See Article on Publisher Site

Abstract

We prove that in dimension $$n \ge 2$$ n ≥ 2 , within the collection of unit-measure cuboids in $$\mathbb {R}^n$$ R n (i.e. domains of the form $$\prod _{i=1}^{n}(0, a_n)$$ ∏ i = 1 n ( 0 , a n ) ), any sequence of minimising domains $$R_k^\mathcal {D}$$ R k D for the Dirichlet eigenvalues $$\lambda _k$$ λ k converges to the unit cube as $$k \rightarrow \infty $$ k → ∞ . Correspondingly we also prove that any sequence of maximising domains $$R_k^\mathcal {N}$$ R k N for the Neumann eigenvalues $$\mu _k$$ μ k within the same collection of domains converges to the unit cube as $$k\rightarrow \infty $$ k → ∞ . For $$n=2$$ n = 2 this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for $$n=3$$ n = 3 was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as $$k \rightarrow \infty $$ k → ∞ . We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first k eigenvalues.

Journal

Integral Equations and Operator TheorySpringer Journals

Published: Nov 1, 2017

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off