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I'm trying to prove the Friedrich's inequality in $\mathbb{R}^n$, but I first want to prove it in $\mathbb{R}$ because using iterated integrals I can then use the same idea in $\mathbb{R}^n$. So let $\Omega$ be an open bounded subset of $\mathbb{R}$.

As $\Omega$ is bounded and $\mathbb{R}$ is well ordered it follows that there exists a least $a\in \mathbb{R}^+$ such that $\Omega \subseteq [-a,a]$. We consider first $v \in C^\infty_0(\Omega)$. Without loss of generality then we can assume that $\Omega$ is connected. Now fix an arbitrary $x\in \Omega$, because $v$ can be taken to be $0$ outside of $\Omega$. By the Cauchy-Schwarz inequality and the fundamental theorem of calculus it follows that \begin{align*} v(x)^2 & =\left (\int_{-a}^xv'(t)dt\right) ^2 \\ & \leq \int_{-a}^{x}(v'(t))^2dt\int_{-a}^{x}dt \\ & \leq \int_{-a}^{x}(v'(t))^2dt\int_{-a}^{a}dt \\ & \leq 2a\int_{\Omega}(v'(x))^2dx. \end{align*} As this is true for any $x\in \Omega$ we can integrate over $\Omega$ on both sides \begin{align*} \int_{\Omega}v(x)^2dx & \leq \int_{\Omega}2a \int_{\Omega}(v'(x'))^2dx'dx \\ & \leq 2a\int_{-a}^a \int_{\Omega}(v'(x'))^2dx'dx\\ & = 4a\int_{\Omega}(v'(x))^2dx. \end{align*} This proves the result for all $v \in C^\infty_0(\Omega)$. For any function $u\in H_0^1(\Omega)$ there exists a sequence of functions $\varphi_n$ in $C^\infty_0$ converging to $u$ in $H^1(\Omega)$. It follows from above that $$ \int_{\Omega}\varphi_n(x)^2dx\leq 4a\int_{\Omega}(\varphi_n'(x))^2dx$$ for all $n\in \mathbb{N}$. As the integral and $t\to t^2$ are continuous functions, and because $\varphi_n \to u$ and $\partial x \varphi_n \to \partial x u$ in $L^2$ it follows that $$\int_{\Omega}u(x)^2dx\leq 4a\int_{\Omega}(u'(x))^2dx.$$

I believe my proof is essentially correct, if not please let me know, but my main issue is in the extension from $C_0^\infty$ to $H_0^1$ using a limiting argument. Am I taking my limits in the right spaces? Am I looking for convergence in $L^2$ or $H^1$ and why? and how can I be more rigorous about it? Any help would be appreciated.

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  • $\begingroup$ Slight nitpick: You go from $2a\int\limits_{-a}^{a}$ to $4a$, when I think you meant $4a^2$. Otherwise it looks fine. BTW, if you just want to show that $\|u\|_{L^2(\Omega)}$ is bounded by some constant times $\|u'\|_{L^2(\Omega)}$, then what you wrote is fine. It's not too hard to refine your proof to show that we can choose this constant to be $\text{diam}(\Omega)$, which is what the version on wikipedia says. $\endgroup$
    – Joey Zou
    Commented Jul 15, 2016 at 2:06
  • $\begingroup$ @JoeyZou yes I do indeed mean $4a^2$ thank you for pointing that out. Yes I thought about using the Lebesgue measure to get the constant to be $\mu(\Omega)^2$ which I think is the same? $\endgroup$
    – K.Power
    Commented Jul 15, 2016 at 8:21

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Yes, your argument is correct. You first prove the inequality on a dense subspace and both sides of the inequality depend continuously (w.r.t. $H^1$) on $v$. Hence, you can pass to the limit via density and obtain the inequality on all of $H_0^1$. This approach also works for many other properties of $H_0^1$.

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