On some nonlinear extensions of the Gagliardo-Nirenberg inequality with applications to nonlinear eigenvalue problems

TL;DR

This paper introduces new nonlinear Gagliardo-Nirenberg type inequalities involving derivatives and transforms, extending classical inequalities and applying them to derive a priori estimates for nonlinear eigenvalue problems.

Contribution

It derives a novel class of nonlinear inequalities that generalize classical second order inequalities and applies them to nonlinear eigenvalue problems.

Findings

New inequalities extend classical Gagliardo-Nirenberg and Oppial inequalities.

Inequalities are used to obtain a priori estimates for nonlinear eigenvalue problems.

The results connect second order inequalities with applications in nonlinear analysis.

Abstract

We derive inequality [\int_{\r} |f^{'}(x)|^ph(f(x))dx \le (\sqrt{p-1})^p\int_{\r}(\sqrt{|f^{"}(x){\cal T}_h(f(x))|})^ph(f(x))dx,] where $f$ belongs locally to Sobolev space $W^{2,1}$ and $f^{'}$ has bounded support. Here $h(...)$ is a given function and ${\cal T}_h(...)$ is its given transform, it is independent of $p$. In case when $h\equiv 1$ we retrieve the well known inequality: (\int_{\r} |f^{'}(x)|^pdx \le (\sqrt{p-1})^p \int_{\r}(\sqrt{|f^{"}(x)f(x)|})^pdx.) Our inequalities have form similar to the classical second order Oppial inequalites. They also extend certain class of inequalities due to Mazya, used to obtain second order isoperimetric inequalities and capacitary estimates. We apply them to obtain new apriori estimates for nonlinear eigenvalue problems.