How to show $\int_{\Omega}|\phi|^2\leq c\sum_{i=1}^{n}\int_{\Omega}|\frac{\partial \phi}{\partial x_i}|^2$

Question

$\int_{\Omega}|\phi|^2\leq c\sum_{i=1}^{n}\int_{\Omega}|\frac{\partial \phi}{\partial x_i}|^2$. Well $\phi\in C_{0}^{\infty}(\Omega)$. So i understant this $\phi(x_1,\ldots,x_n)=\int_{-a}^{x_1}\frac{\partial \phi}{\partial x_1}(\rho_1,x_2,\ldots,x_n)d\rho_1$ and $|\int_{-a}^{x_1}\frac{\partial \phi}{\partial x_1}(\rho_1,x_2,\ldots,x_n)d\rho_1|=|\int_{-a}^{x_1}1\frac{\partial \phi}{\partial x_1}(\rho_1,x_2,\ldots,x_n)d\rho_1|\leq \int_{-a}^{a}|1\frac{\partial \phi}{\partial x_1}(\rho_1,x_2,\ldots,x_n)|d\rho_1\leq (\int_{-a}^{a}1^2d\rho_1)^{\frac{1}{2}}(\int_{-a}^{a}|\frac{\partial \phi}{\partial x_1}(\rho_1,x_2,\ldots,x_n)|^2d\rho_1)^{\frac{1}{2}}=(2a)^{\frac{1}{2}}(\int_{-a}^{a}|\frac{\partial \phi}{\partial x_1}(\rho_1,x_2,\ldots,x_n)|^2d\rho_1)^{\frac{1}{2}}$ (im using $|x_i|<a$)imply $|\phi(x_1,\ldots,x_n)|^2\leq 2a \int_{-a}^{a}|\frac{\partial \phi}{\partial x_1}(\rho_1,x_2,\ldots,x_n)|^2d\rho_1$. After that we integrate respect to $x_1$ in both side from $-a$ to $a$ i mean $\int_{-a}^{a}|\phi|^2dx_1\leq 4a^2\int_{-a}^{a}|\frac{\partial \phi}{\partial x_1}(\rho_1,x_2,\ldots,x_n)|^2d\rho_1$ to here Iam ok, but if i continued integrating respect to $x_2, \ldots, x_n$ i think that i will obtain $\int_{(-a,a)^n} |\phi|^2dx \leq (2a)^n \int_{-a}^{a}|\frac{\partial \phi}{\partial x_1}|^2 d\rho_1$ is not? How to obtain the sum in the statement? I would to like understand how to finish the statement, i will appreciate any hints or help!! Thanks

Answer 1

This is called Poincaré's inequality and not true for general domains. For instance, it is false for $\Omega = \mathbb R^n$: Just take a nontrivial smooth $\phi$ with compact support and consider $\phi_\lambda(x) := \phi(\lambda x)$ for $\lambda > 0$.

Note that $c$ should not depend on $\phi$ else the inequality is trivial. However, this means $c$ can also not depend on $\mathrm{supp}\, \varphi$ and if I understand you correctly, your $a$ is $\inf \{\,a' \in \mathbb R : \exists x \in \mathbb R^{n-1} : (a', x) \in \mathrm{supp}\,\phi\,\}$. Long story short, $c$ may not depend on your $a$ and I guess that is a problem you will run into.

The inequality however holds for instance for domains bounded in one direction. Then one can argue quite similar as you do and in the end estimate \begin{align*}\int_{\Omega'} \int_{-a}^a |\partial_1 \phi|^2 \mathrm d \rho_1 \mathrm d \rho' &= \int_\Omega |\partial_1 \phi|^2 \le \sum_{i=1}^n \int_\Omega |\partial_i \phi|^2. \end{align*}