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

$$\int_{\Omega}|\phi|^2\leq c\sum_{i=1}^{n}\int_{\Omega}|\frac{\partial \phi}{\partial x_i}|^2$$. 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 $$|x_i|)imply $$|\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

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*}
• Thank you, i forget to say $\Omega$ is bounded, and what happen if dont have that is bounded in one direction? Commented Jul 5, 2020 at 2:32
• As I said, the inequality does not hold for all domains, for instance it is false for $\Omega = \mathbb R^n$. However, “bounded in one direction” is not a necessary conditions. However, determining the optimal condition is out of scope of this questions, I guess.