English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In the 19th century, Lord Rayleigh conjectured that among all clamped plateswith given area, the disk minimizes the fundamental tone. In the 1990s, N. S.Nadirashvili proved the conjecture in $\mathbb{R}^2$ and M. S. Ashbaugh und R.D. Benguria gave a proof in $\mathbb{R}^2$ and $\mathbb{R}^3$.  In the present paper, we prove existence of an optimal domain for minimizingthe fundamental tone among all open and bounded subsets of $\mathbb{R}^n$,$n\geq 4$, with given measure. We formulate the minimization of the fundamentaltone of a clamped plate as a free boundary value problem with a penalizationterm for the volume constraint. As the penalization parameter becomes small weshow that the optimal shape problem is solved.

Existence of an optimal domain for minimizing the fundamental tone of a clamped plate of prescribed volume in arbitrary dimension