Topological Methods in Nonlinear Analysis

On certain variant of strongly nonlinear multidimensional interpolation inequality

Tomasz Choczewski and Agnieszka Kałamajska

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Abstract

We obtain the inequality \[ \int_{\Omega}|\nabla u(x)|^ph(u(x))\,dx \leq C(n,p)\int_{\Omega} \Big( \sqrt{ |\nabla^{(2)} u(x)| |{\mathcal T}_{h,C}(u(x))|}\Big)^{p}h(u(x))\,dx, \] where $\Omega\subset \mathbb R^n$ and $n\ge 2$, $u\colon\Omega\rightarrow \mathbb R$ is in certain subset in second order Sobolev space $W^{2,1}_{\rm loc}(\Omega)$, $\nabla^{(2)} u$ is the Hessian matrix of $u$, ${\mathcal T}_{h,C}(u)$ is a certain transformation of the continuous function $h(\,\cdot\,)$. Such inequality is the generalization of a similar inequality holding in one dimension, obtained earlier by second author and Peszek.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 1 (2018), 49-67.

Dates
First available in Project Euclid: 24 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1524535226

Digital Object Identifier
doi:10.12775/TMNA.2017.050

Mathematical Reviews number (MathSciNet)
MR3867979

Zentralblatt MATH identifier
07029861

Citation

Choczewski, Tomasz; Kałamajska, Agnieszka. On certain variant of strongly nonlinear multidimensional interpolation inequality. Topol. Methods Nonlinear Anal. 52 (2018), no. 1, 49--67. doi:10.12775/TMNA.2017.050. https://projecteuclid.org/euclid.tmna/1524535226


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