----- > [!proposition] Proposition. ([[Gagliardo-Nirenberg's Inequality in two dimensions]]) > One has $\|u\|_{L^{2}(\mathbb{R}^{2})}^{2} \leq \|\partial_{1}u\|_{L^{1}(\mathbb{R}^{2})} \|\partial_{2}u\|_{L^{1}(\mathbb{R}^{2} )} \text{ for all } u \in C^{\infty}_{c}(\mathbb{R}^{2}).$ ^proposition > [!proposition] Corollary. (Ladyzhenskaya's Inequality) > By applying the above to $v^{2}$ and using the [[Cauchy-Schwarz Inequality]], we obtain $\|v\| ^{2}_{L^{4}(\mathbb{R}^{2})} \leq \sqrt{ 2 } \|v\|_{L^{2}(\mathbb{R}^{2})}\|Dv\|_{L^{2}(\mathbb{R}^{2})} \text{ for all } v \in C^{\infty}_{c}(\mathbb{R}^{2}).$ ^proposition > [!proof] Proof of Corollary. > Let $v \in C^{\infty}_{c}(\mathbb{R}^{2})$ be arbitrary. Have > > $\begin{align} > (\|v\|^{2}_{4})^{2} &= \|v^{2}\|_{2}^{2} \\ > &\leq { \|\partial_{1}(v^{2})\|_{1} \, \|\partial_{2}(v^{2})\|_{1} } \ \ \ \ (\mathrm{i}) \\ > &= { \left( \int _{\mathbb{R}^{2}} |2 v\partial_{ 1}v| \right) \left( \int _{\mathbb{R}^{2}} |2v \partial_{2}v| \, \right) } \\ > &= { \|2v\|_{2}^{2} \|\partial_{1}v\|_{2}^{} \|\partial_{2}v\|_{2}^{} } \ \ \ \ \text{(Cauchy-Schwarz)} \\ > & \leq \|2v\|_{2}^{2} \|Dv\|_{2}^{2}, > \end{align}$ > where the last step uses $\|\partial_{i}v\|_{2} \leq \|Dv\|_{2}$. > Taking the square root on both sides yields $\|v\|_{4}^{2} \leq \sqrt{ 2 } \|v\|_{2} \|Dv\|_{2}$ > as desired. ^54bfc2 > [!proof]- Proof. ([[Gagliardo-Nirenberg's Inequality in two dimensions]]) > We use use a strategy similar to the proof of [[Gagliardo-Nirenberg-Sobolev Inequality|GNS]] given in lecture. > Define maps $f_{x_{1}}:\mathbb{R} \to \mathbb{R}$, $f_{x_{1}}(x_{2})=u(x_{1},x_{2})$ and similarly $f_{x_{2}}(x_{1})=u(x_{1},x_{2})$. By [[The Fundamental Theorem of Calculus]] (applicable because of [[compact]] [[support]]), $u(x_{1},x_{2})=f_{x_{2}}(x_{1})=\int _{-\infty}^{x_{1}} \partial_{1}u(y_{1},x_{2}) \, dy_{1}.$ > Thus $\begin{align} > |u(x_{1},x_{2})|& \leq \int _{-\infty}^{x_{1}} | \partial_{1}u(y_{1},x_{2})| \, dy_{1} \\ > & \leq \int _{-\infty} ^{\infty} |\partial_{1}u(y_{1}, x_{2})| \, dy_{1}. > \end{align}$Symmetrically, $|u(x_{1},x_{2})| \leq \int _{-\infty}^{\infty} |\partial_{2}u(x_{1},y_{2})| \, dy_{2}.$ > Now $\begin{align} > |u(x_{1},x_{2})|^{2} & \leq \left( \int _{-\infty} ^{\infty} |\partial_{1}u(y_{1}, x_{2})| \, dy_{1} \right)\left( \int _{-\infty}^{\infty} |\partial_{2}u(x_{1},y_{2})| \, dy_{2} \right), > \end{align}$ > thus $\begin{align} > \|u\|_{2}^{2} &= \int _{\mathbb{R}^{2}} |u|^{2}\, \\ > &= \int _{-\infty}^{\infty} \int _{-\infty}^{\infty} |u(x_{1},x_{2})|^{2} \, dx_{1} \, dx_{2} \\ > &\leq \int _{-\infty}^{\infty} \int _{-\infty}^{\infty} \left( \int _{-\infty} ^{\infty} |\partial_{1}u(y_{1}, x_{2})| \, dy_{1} \right) \left( \int _{-\infty}^{\infty} |\partial_{2}u(x_{1},y_{2})| \, dy_{2} \right) \,dx_{1} \, dx_{2} \\ > &= \left( \int _{-\infty} ^{\infty} \int _{-\infty} ^{\infty} |\partial_{1}u(y_{1},x_{2})| \, dy_{1} \, dx_{2} \right) \int _{-\infty} ^{\infty} \int _{-\infty}^{\infty} |\partial_{2}u(x_{1},y_{2})| \, dy_{2}\, dx_{1} \\ > &= \underbrace{ \left( \int _{\mathbb{R}^{2}}| \partial_{1} u | \right) }_{ \|\partial_{1}u\|_{1} } \underbrace{ \left( \int _{\mathbb{R}^{2}} | \partial_{2}u | \right) }_{ \|\partial_{2}u\|_{1} }. > \end{align}$ > > where the penultimate step used [[Fubini's Theorem]]. The inequality follows. ^33316d ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` [[derivative]]