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arXiv 1104.1967math.AP
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Gagliardo-Nirenberg不等式的非线性扩展及其在非线性本征值问题中的应用

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

Agnieszka Kałamajska, Jan Peszek

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30 pages
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摘要：

我们得到不等式[\int_{\r}| f ^n '}（x）|^ph（f（x））dx \le（\sqrt{p-1}）^p \int_{\r}（\sqrt{|f^{"}(x){\cal T}_h(f(x))|}）^ph（f（x））dx，]其中$f$局部属于Sobolev空间$W^{2,1}$，并且$f^{'}$具有有界支撑。这里$h(...)$是一个给定的函数，而${\cal T}_h(...)$是它的给定变换，它与$p$无关。在$h\equiv 1$的情况下，我们检索众所周知的不等式：（\int_{\r}| f ^n '}（x）|^pdx \le（\sqrt{p-1}）^p \int_{\r}（\sqrt{|f^{"}(x)f(x)|}）^pdx.）我们的不等式的形式类似于经典的二阶Oppial不等式。他们还推广了Mazya的某些类不等式，用于获得二阶等周不等式和容量估计。应用它们得到了非线性本征值问题的新的先验估计。

英文摘要：

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.

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