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The following is the beginning of a proof of the Poincaré inequality:

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Here are my questions:

  • How is the limiting argument made so that "it is sufficient to prove the inequality for $u\in C_c^\infty(\Omega)$?
  • What does "The inequality is invariant under rotations and translations" mean? Could anyone write it out explicitly?
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  • $\begingroup$ For the first question because it still applies by approximation, taking the limit under the integral sign. For the second question, I think which the reference is to the Lebesgue measure. $\endgroup$ – Andrew Sep 5 '16 at 18:12
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    $\begingroup$ Id say the "proving it for $u \in C^\infty_c$ is enough" part is the most important idea of all functional analysis. $\endgroup$ – reuns Sep 5 '16 at 20:36
  • $\begingroup$ @user1952009 You really are right! in every book I find ... is sufficient to prove that..ok, why? $\endgroup$ – Andrew Sep 5 '16 at 21:21
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1) Suppose you prove the inequality for all smooth $u$ with compact support in $\Omega$. Let $u \in H^1_0(\Omega)$. Then by density of $C_c^\infty(\Omega)$ in $H_0^1$, there exists a sequence $\phi_k \in C_c^\infty(\Omega)$ converging to $u$ in the $H^1_0(\Omega)$ norm. Therefore $$\lim_{k\to \infty} \int_\Omega \phi_k^2 \, dx = \int_\Omega u^2 \, dx \ \ \ \text{ and } \ \ \ \lim_{k\to \infty} \int_\Omega |D\phi_k|^2 \, dx = \int_\Omega |Du|^2 \, dx.$$ Since the estimate holds for each term $\phi_k$ in the sequence, with the same uniform constant $C$, it must hold for $u$ as well. This is an argument by density or approximation.

2) A rotation and translation is a transformation of the form $y=Ax + b$, where $b \in \mathbb{R}^n$ and $A$ is an orthogonal matrix ($A^TA=AA^T=I$, and $\text{det}(A)=1$). Suppose we change variables and set $v(y)=u(x)=u(A^T(y-b))$. Then $dy = |\text{det}(A)|dx = dx$ and $Dv(y) = ADu(x)$. Since $A$ is orthogonal, $|Dv(y)|=|Du(x)|$ and we have $$\int_\Omega u(x)^2 \,dx = \int_{\Omega'} v(y)^2 \, dy \ \ \ \text{ and } \ \ \ \int_\Omega |Du(x)|^2 \,dx = \int_{\Omega'} |Dv(y)|^2 \, dy,$$ where $\Omega' = A\Omega + b$. Since both sides of the estimate are invariant under rotations and translations, we may as well assume the bounded direction lies in the $x_n$ direction between $0 < x_n < a$.

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