Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Hardy-type inequality in variable exponent Lebesgue spaces derived from nonlinear problem (1407.6226v3)

Published 23 Jul 2014 in math.AP

Abstract: We derive a family of weighted Hardy-type inequalities in the variable exponent 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 Lipschitz function. The involved measures depend on a certain solution to the partial differential inequality involving $p(x)$-Laplacian ${-}\Delta_{p(x)} u\geqslant \Phi$, where $\Phi$ is a given locally integrable function, and $u$ is defined on an open and not necessarily bounded subset $\Omega\subseteq\mathbb{R}n $, and a certain parameter $\beta$. We derive new Caccioppoli-type inequality for the solution $u$. As its consequence we get Hardy-type inequality. We illustrate the result by several one-dimensional examples.

Summary

We haven't generated a summary for this paper yet.