# A problem in PDE .

I need help for the solution of this problem : For $u:\Omega \to R$, $h>0$ and $i\in {1,\ldots,d}$, we define the difference quotient

$D^h_i u(x)= ({u(x+he_i)-u(x))/h}$ on $\Omega_0=\Omega_0(h,i)={x\in\Omega: x+he_i\in \Omega }$. Here $e_i$ denotes the $i^{th}$ standard basis vector . I want to show the following :

a) For all $i\in {1,\ldots,d}$, $h>0$, and $u \in H^1(\Omega)$, $||D^h_iu||_{L^2(\Omega_0)} \le ||\partial_iu||_{L^2(\Omega)}$.

B) Suppose that for every $V$ compactly contained in $\Omega$ and every $h>0$ such that $D^h_iu$ is defined on $V$, we have $||D^h_iu||_{L^2(V)} \le A$ for some $A \ge 0$, show that the $i^{th}$ weak partial derivative $\partial_iu$ exists and satisfies $||\partial_i u||_{L^2(\Omega)} \le A$. I don't even have any idea to start this problem . Any kind of help would be great . Thanks

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What do we know about $\Omega$? – Davide Giraudo Jun 3 '12 at 9:07
a) is lemma 7.23, b) lemma 7.24 in Gilbarg and Trudingers "Elliptic Partial Differential Equations of Second Order". – user20266 Jun 3 '12 at 9:08
@Davide Giraudo : Not much of info is given about $\Omega$. – Theorem Jun 3 '12 at 9:24