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 a family of weighted Hardy-type inequalities in the variableexponent Lebesgue space with an additional term of the form  \[ \int_\Omega\ |\xi|^{p(x)} \mu_{1,\beta}(dx)\leqslant \int_\Omega |\nabla\xi|^{p(x)}\mu_{2,\beta}(dx)+\int_\Omega \left|\xi{\log \xi} \right|^{p(x)}\mu_{3,\beta}(dx), \] where $\xi$ is any compactly supported Lipschitzfunction. The involved measures depend on a certain solution to the partialdifferential inequality involving $p(x)$-Laplacian ${-}\Delta_{p(x)} u\geqslant\Phi$, where $\Phi$ is a given locally integrable function, and $u$ is definedon an open and not necessarily bounded subset $\Omega\subseteq\mathbb{R}^n $,and a certain parameter $\beta$. We derive new Caccioppoli-type inequality forthe solution $u$. As its consequence we get Hardy-type inequality. Weillustrate the result by several one-dimensional examples.

Hardy-type inequality in variable exponent Lebesgue spaces derived from nonlinear problem