# PDE Regularity Theory - Brezis

The following lemma is from Brezis' text. Throughout we take $$\Omega=\mathbb{R}^{N}_{+}$$.

Let $$u\in H^{2}(\Omega)\cap H^{1}_{0}(\Omega)$$ satisfying, \begin{align} \int\nabla u\cdot\nabla\varphi+\int u\varphi=\int f\varphi\quad\forall \varphi\in H^{1}_{0}(\Omega). \end{align} Then $$Du\in H^{1}_{0}(\Omega)$$ and, moreover, \begin{align} \int\nabla(Du)\cdot\nabla\varphi+\int(Du)\varphi=\int f\varphi\quad\forall\varphi\in H^{1}_{0}(\Omega). \end{align}

The proof is as follows:

Let $$h=|h|e_{j}$$, $$1\leq j\leq N-1$$, so that $$D_{h}u\in H^{1}_{0}$$. By Lemma 9.6 we have, \begin{align} \|D_{h}u\|_{H^{1}}\leq\|u\|_{H^{2}}. \end{align} Thus there exists a sequence $$h_{n}\rightarrow 0$$ such that $$D_{h_{n}}u$$ converges weakly to some $$g$$ in $$H^{1}_{0}$$. In particular, $$D_{h_{n}}u\rightharpoonup g$$ weakly in $$L^{2}$$. For $$\varphi\in C_{c}^{\infty}(\Omega)$$ we have, \begin{align} \int(D_{h}u)\varphi=-\int u(D_{-h}\varphi), \end{align} and in the limit, as $$h_{n}\rightarrow 0$$, we obtain, \begin{align} \int g\varphi=-\int u\frac{\partial\varphi}{\partial x_{j}}\quad\forall\varphi\in C_{c}^{\infty}(\Omega). \end{align} Therefore, $$\frac{\partial u}{\partial x_{j}}=g\in H^{1}_{0}(\Omega)$$.

My Question: Why does a bound on the $$H^{1}$$ norm of $$D_{h}u$$ guarantee a weakly convergent sequence $$D_{h_{n}}u$$? Also in the last part where he is taking the limit $$h_{n}\rightarrow 0$$ in the integrals, there are no $$h_{n}$$ terms appearing, so what are we really doing here?

Why does a bound on the $$H^1$$ norm of $$D_hu$$ guarantee a weakly convergent sequence $$D_{h_n}u$$?
Since this is also a boundedness of $$D_{h_n} u$$. Take a sequence $$h_n=|h_n|e_j$$ and now let $$|h_n| \to 0$$. Hence also $$h_n \to 0$$ and $$h_n$$ corresponds to a tangential translation operator i.e. on the same lines from the proof we get $$\|D_{h_n} u\|_{H^1}$$ bounded.
Also in the last part where he is taking the limit $$h_n→0$$ in the integrals, there are no $$h_n$$ terms appearing, so what are we really doing here?
He writes down the formula for a general $$h$$, yes, but he applies the limit to the equation where we replace $$h$$ by $$h_n$$, i.e.
$$\int g \varphi \leftarrow \int (D_{h_n} u) \varphi = \int u (D_{-h_n} \varphi) \rightarrow -\int u \partial_{x_j}\varphi.$$
To add more explanations for your first question, take the sequence of function $$v_n = D_{h_n}u$$. Since $$\|v_n\|_{H^1} ≤ \|u\|_{H^2}$$ the sequence $$v_n$$ is bounded in $$H^1_0$$ uniformly in $$n$$ and thus converges weakly in $$H^1_0$$ so some function $$g$$ (by Banach-Alaoglu theorem, i.e. the weak compactness of bounded sets).