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 Tools annual

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 Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

The Poincare Inequality is an extremely useful tool in the analysis of PDEs.A significant amount of literature has dealt with finding the optimal constant$C(p,\Omega)$, depending only the domain $\Omega$, and the $L^p$ norm inquestion. For convex subsets $\Omega\subset \mathbb{R}^n$, an optimal constantis known. There has also been some work done for $\textit{specific}$ non-convexsubsets. For a general open, bounded subset, finding the optimal constant is anon-trivial task. This paper will outline a method for finding a non-trivialestimate for $C$ when $\Omega\subset \mathbb{R}^n$ is non-convex, by firstobserving $\Omega\subset \mathbb{R}^n$ has finite Lebesgue measure, and usinglocal estimates on subsets of $\Omega$ to obtain a global estimate on $\Omega$.

Non-trivial Poincare Constant in any Subset of $\mathbb{R}^n$