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

We derive upper bounds for the trace of the heat kernel  Z ( t ) of the Dirichlet Laplace operator in an open set Ω ⊂ ℝ  d  , with  d  ⩾ 2. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on  Z ( t ) in domains of infinite volume. For domains of finite volume the bound on  Z ( t ) decays exponentially as  t  tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short‐time asymptotics of  Z ( t ). To prove the result, we employ refined Berezin–Li–Yau inequalities for eigenvalue means.

Universal bounds for traces of the Dirichlet Laplace operator