Prove Friedrichs' inequality

I’m trying to show that the theorem (Friedrichs’ inequality) in my book:

Assume that $\Omega$ be a bounded domain of Euclidean space $\Bbb R^n$. Suppose that $u: \Omega \to \Bbb R$ lies in the Sobolev space $W_{0}^{1,p}(\Omega)$. Then $\exists$ const $C_{\Omega}$ such that
$$\|u\|_{L_p(\Omega)} \le C_{\Omega} \left(\int_\Omega\sum_{j=1}^{n}\left| \dfrac{\partial u}{\partial x_j}\right|^p \mathrm{d}x \right)^{\dfrac{1}{p}}$$

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I tried to use $\|u\|_{L_p(\Omega)}=\left(\int_\Omega\left |u(x)\right |^p\rm{dx} \right)^{\frac{1}{p}}$.
But I have no solution. Can anyone help me?

Any help will be appreciated! Thanks!

Solutions Collecting From Web of "Prove Friedrichs' inequality"

Assume that $\Omega$ lies between the planes $x_n=0$ and $x_n=a>0$. We first consider the case where $u\in C_0^\infty (\Omega)$. Write $x=(x’,x_n)$ with $x’=(x_1,…,x_{n-1})$ and note that $$|u(x’,x_n)|=|u(x’,x_n)-u(x’,0)|=\left|\int_0^{x_n} \frac{d}{dt}u(x’,t)dt\right|\tag{1}$$

Hint 1: Apply Holder inequality to $(1)$ in order to get the desired inequality for functions in $C_0^\infty(\Omega)$.

Hint 2: Remember the definition of $W_0^{1,p}(\Omega)$.

Edit: You have reached this part (after fixing “my” typos) $$\left \| u \right \|_{L_p(\Omega)}^{p}=\int_{\Omega}\left | u(x) \right |^p \rm d x \le \int_{\Omega} \left ( x_n^{\frac{p}{q}}\cdot \left ( \int_{0}^{x_n} \left|\frac{\partial }{\partial t}u(x’,t)\right|^p \rm d t\right ) \right ) \rm dx $$

Now, note that if $x_n\leq a$, then

$$\int_{\Omega} \left ( x_n^{\frac{p}{q}}\cdot \left ( \int_{0}^{x_n} \left|\frac{\partial }{\partial t}u(x’,t)\right|^p \rm d t\right ) \right ) \rm dx\leq \int_{\Omega} \left ( x_n^{\frac{p}{q}}\cdot \left ( \int_{0}^{a} \left|\frac{\partial }{\partial t}u(x’,t)\right|^p \rm d t\right ) \right ) \rm dx$$

Now you have to do some “tricks” in order to apply Fubini’s Theorem. For example, because $u\in C_0^\infty (\Omega)$, you can consider (without losse of generality, why?) that $$\int_{\Omega} \left ( x_n^{\frac{p}{q}}\cdot \left ( \int_{0}^{a} \left|\frac{\partial }{\partial t}u(x’,t)\right|^p \rm d t\right ) \right ) \rm dx=\int_{\mathbb{R}^{n-1}\times (0,a)} \left ( x_n^{\frac{p}{q}}\cdot \left ( \int_{0}^{a} \left|\frac{\partial }{\partial t}u(x’,t)\right|^p \rm d t\right ) \right ) \rm dx’ dx_n\tag{2}$$

We apply Fubini’s Theorem to the right hand side of $(2)$ to conclude that

$$\int_{\mathbb{R}^{n-1}\times (0,a)} \left ( x_n^{\frac{p}{q}}\cdot \left ( \int_{0}^{a} \left|\frac{\partial }{\partial t}u(x’,t)\right|^p \rm d t\right ) \right ) \rm dx’ dx_n=\left(\int_0^a x_n^{\frac{p}{q}}dx_n \right)\int_\Omega \left|\frac{\partial }{\partial t}u(x’,t)\right|^p \rm d t \rm dx’\tag{3}$$

We obtain from $(3)$ that $$\|u\|_p\leq \left(\int_0^a x_n^{\frac{p}{q}}dx_n \right)^{1/p}\left\|\frac{\partial u}{\partial x_n}\right\|_p$$

Can you conclude now?