Tomasz Choczewski | Searcharxiv

arXiv2018

On one variant of strongly nonlinear Gagliardo-Nirenberg inequality involving Laplace operator with application to nonlinear elliptic problems

We obtain the inequality $$\int_Ω|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_Ω \left( \sqrt{ |Δu(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $Ω\subset \mathbf{R}^n$ is a bounded Lipschitz domain, $u\in W^{2,1}_{loc}(Ω)$ is postive and obeys some additional assumptions, $Δu$ is the Laplace operator, ${\cal T}_{h,C}(\cdot )$ is certain transformation of the continuous function $h(\cdot)$. We also explain how to apply such inequality to deduce regularity for solutions of nonlinear eigenvalue problems of elliptic type for degenerated PDEs, with the illustration within the model of electrostatic micromechanical systems (MEMS).

math.AP↗

arXiv2016

On certain variant of strongly nonlinear interpolation inequality in dimension n

We obtain the inequality $$\int_Ω|\nabla u(x)|^ph(u(x))dx\leq C(n,p)\int_Ω \left( \sqrt{ |\nabla^{(2)} u(x)||{\cal T}_{h,C}(u(x))|}\right)^{p}h(u(x))dx,$$ where $Ω\subseteq {\bf R}^n$ and $n\ge 2$, $u:Ω\rightarrow {\bf R}$ is in certain subset in second order Sobolev space $W^{2,1}_{loc}(Ω)$, $\nabla^{(2)} u$ is the Hessian matrix of $u$, ${\cal T}_{h,C}(u)$ is certain transformation of the continuous function $h(\cdot)$. Such inequality is the generalization of similar inequality holding in one dimension, obtained earlier by second author and Peszek.

math.AP↗