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My question:

Let $\Omega$ in $\mathbb{R}^n$ bounded. For all $\varepsilon>0$, there exists a constant $C(\varepsilon)>0$ such that $$ \label{lemma_gagliardo_nirenberg_2} \|\varphi\|_{L^2(\Omega)}^2 \le \varepsilon \|\nabla \varphi\|_{L^2(\Omega)}^2 + C(\varepsilon) \|\varphi\|_{L^1(\Omega)}^2 \quad\text{ for all }\varphi \in W^{1,2}(\Omega). $$

mathmatix
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1 Answers1

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From Gagliardo-Nirenberg it holds that $$ \|u\|^2_{L^2} \leq C_1 \|\nabla u\|_{L^2}^{2n/(n+2)} \|u\|_{L^1}^{4/(n+2)} + C_2 \|u\|^2_{L^1}. $$ Now you can apply Young's inequality $$ a b \leq \frac{\varepsilon}{C_1} a^p + C(\varepsilon) b^{p'}, \qquad a,b>0, $$ with $p = \frac{n+2}{n}$ (and $p' = \frac{n+2}{2}$).

Rigel
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  • Do you have a reference for this form of the Gagliardo-Nirenberg inequality? And does this hold on a bounded domain? Because I only know the inequality with $+ C_2 |\varphi|_{L^1(\Omega)}$ on the r.h.s. – mathmatix Jul 31 '20 at 13:42
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    You can see the paper by Brezis and Mironescu: https://hal.archives-ouvertes.fr/hal-01626613/document/ – Rigel Jul 31 '20 at 13:45
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    @mathmatix: anyway, you can use the standard Gagliardo-Nirenberg inequality with the additional term (see my edited answer). – Rigel Jul 31 '20 at 13:54
  • Recall that $(A+B)^2 \leq 2(A^2 + B^2)$. – Rigel Jul 31 '20 at 14:29