I started a study on the Navier Stokes equations and my supervisor gave me a first result to work on which is the J.L. Lions Lemma which basically states the following:
If $f\in H^{-1}(\Omega)$ and $\nabla f\in H^{-1}(\Omega)$. Then $f\in L^2(\Omega)$
My supervisor told me that since $\Omega$ is a Lipschitz domain and from the proof of the Global approximation by functions smooth up to the boundary theorem and its Evans's Partial Differential Equations:
making the convolution of $f\in H^{-1}(\Omega)\, \nabla f\in H^{-1}(\Omega)$, we are in $L^2$. We can then use the Necas inequality.
My ideas:
- I think that the step 1 and 2 of proof is for build the open covers $\{V_i\}_{i=0}^{q}$ where $V_i=\Omega\cap B(x_0,\frac{r_i}{2})$ and we can choose $\Omega = \cup_{i=0}^q V_i$.
- We can also take a classic regularizing sequence $(\rho_n)$ for $\epsilon>0$, let $\eta_\epsilon = \rho_{[1/\epsilon]+1}$ or again $f_\epsilon(x) = f(x^\epsilon) = f(x+\lambda\epsilon e_n)$ and I can take $$g_\epsilon = \eta_\epsilon * f_\epsilon.$$ I think it must be proven that $g_\epsilon \in L^2(V_i)$ and with that we can later use the Necas' inequality
- Then, we seek to provide hypotheses to ensure the existence of the unit partition $\theta_0,\,\theta_2\,...\,\theta_q$ associated with the open cover $\{V_i\}$ and we take $h = \theta_i g_\epsilon$ and then prove that (by Necas' inequality) $$\|h\|_{L^2(\Omega)} = \|h\|_{L^2(V_i)}\leq C(\|g_\epsilon\|_{H^{-1}(\Omega)}+\|\nabla g_\epsilon\|_{H^{-1}(\Omega)}),$$ and passing to the limit when $\epsilon \rightarrow 0 $ we would get $$\|f\|_{L^2(\Omega)}\leq C(\|f\|_{H^{-1}(\Omega)}+\|\nabla f\|_{H^{-1}(\Omega)})<\infty.$$ We can conclude that $f\in L^2(\Omega)$
Any indications or suggestions are welcome
The approximation argument is a lot simpler than you might think; for negative order spaces the zero extension $H^{-1}(\Omega) \hookrightarrow H^{-1}(\Bbb R^n)$ is continuous, so you you can simply extend by zero and mollify.
The key ingredient is the estimate $$ \lVert g \rVert_{L^2(\Omega)} \leq C \left( \lVert g \rVert_{H^{-1}(\Omega)} + \lVert \nabla g \rVert_{H^{-1}(\Omega)}\right) $$ you mention in the comments, which is valid for $g \in L^2(\Omega).$ This is where one needs $\Omega$ to be a Lipschitz domain, as the proof I involves using Fourier analysis in the full space and the continuity of the restriction $H^{-1}(\Bbb R^n) \to H^{-1}(\Omega),$ and the latter is where one needs some regularity of the boundary. From here the strategy as you noted is to regularise $f,$ apply the above, and conclude by passing to the limit. For this we need the following.
Lemma: If $g \in H^{-1}(\Omega),$ extending by zero let $g_{\varepsilon} = g \ast \rho_{\varepsilon}$ where $\rho_{\varepsilon}$ is a standard mollifier. Then $g_{\varepsilon} \in C^{\infty}(\overline\Omega)$ and $g_{\varepsilon} \to g$ in $H^{-1}(\Omega).$
Proof: Since $g \in H^{-1}(\Omega),$ there exists $g_0,g_1,\dots,g_n \in L^2(\Omega)$ such that $g = g_0 + \sum_{i=1}^n D^ig_i.$ Then extending each $g_i$ by zero to $\Bbb R^n$ we have $$ g_{\varepsilon} = g_0 \ast \rho_{\varepsilon} + \sum_{i=1}^n D^i(g_i \ast \rho_{\varepsilon}) $$ which is smooth on $\overline\Omega.$ Now we can estimate $$ \lVert g-g_{\varepsilon} \rVert_{H^{-1}(\Omega)} \leq \sum_{i=0}^n \lVert g_i \ast \rho_{\varepsilon} - g_i \rVert_{L^2(\Omega)} \to 0 $$ as $\varepsilon \to 0,$ which proves the lemma.
Now the rest is straightforward; since $f_{\varepsilon} = f \ast \rho_{\varepsilon} \in C^{\infty}(\overline\Omega) \subset L^2(\Omega)$ we can apply Necas' inequality to get \begin{align*} \liminf_{\varepsilon \to \infty}\lVert f_{\varepsilon} \rVert_{L^2(\Omega)} &\leq \lim_{\varepsilon \to \infty}C \left( \lVert f_{\varepsilon} \rVert_{H^{-1}(\Omega)} + \lVert \nabla f_{\varepsilon} \rVert_{H^{-1}(\Omega)}\right) \\ & = C \left( \lVert f \rVert_{H^{-1}(\Omega)} + \lVert \nabla f \rVert_{H^{-1}(\Omega)}\right) < \infty, \end{align*} so by Fatou's lemma we conclude that $f \in L^2(\Omega).$