Hello everyone (sorry for the messy title!) ! I am currently working with Gilbarg & Trudinger, Elliptic Partial Differential Equations of Second Order (so I use their notation and definition). I am trying to prove the following :
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $u \in \mathcal{C}^2(\Omega)\cap \mathcal{C}^0(\overline{\Omega})$, $f \in \mathcal{C}^\alpha(\Omega)$ functions such that $\Delta u = f$ and $u\vert_{\delta \Omega} = \psi$, where $\psi : \delta \Omega \longrightarrow \mathbb{R}$ is a continuous function. Show that there exists a constant $C=C(n,\alpha,diam(\Omega))$ such that : \begin{align*} |u|^*_{2;\alpha;\Omega} \leq C(\sup_{\delta \Omega} |\psi| + |f|^{(2)}_{0;\alpha;\Omega}) \end{align*}
Here is the definition of the (semi-)norms I am using, for $u \in \mathcal{C}^{k,\alpha}(\Omega)$ :
Well, here is my proof, but I am not sure about it so it would be nice if someone could review it.
We proved in Gilbarg & Trudinger (Theorem 4.8), that for an open set of $\mathbb{R}^n$, and for $u \in \mathcal{C}^2(\Omega), f \in \mathcal{C}^\alpha(\Omega)$ satisfying $\Delta u = f$, then the inequality $|u|^*_{2;\alpha;\Omega} \leq C(|u|_{0,\Omega} + |f|^{(2)}_{0;\alpha;\Omega})$ holds.
In my exercice, we are working with a bounded domain. The above inequality holds, by the theorem 4.8. I am then using an apriori bounds to say that in fact, we also have the followng inequality :
\begin{align*} \sup_{\Omega}|u| \leq C(\sup_{\delta \Omega} |\psi| + C' \sup_{\Omega}(\frac{|f|}{\lambda}))\end{align*}*
where $C'$ is a constant which depends only on $diam(\Omega)$, which is finite. $\lambda$ is in fact equal to $1$ here, as the smallest eigenvalue of $[a_{i,j}(x)]$ (in the definition of an elliptic operator).
Then, as $|u|_{0,\Omega} = \sup_{\Omega} |u|$, the following inequality holds :
\begin{align*} |u|^*_{2;\alpha;\Omega} \leq C(\sup_{\delta \Omega} |\psi| + C' \sup_{\Omega}|f| + |f|^{(2)}_{0;\alpha;\Omega}) \end{align*}
and so, by definition of $|f|^{(2)}_{0;\alpha;\Omega}$ :
\begin{align*} |u|^*_{2;\alpha;\Omega} \leq C''(\sup_{\delta \Omega} |\psi| + |f|^{(2)}_{0;\alpha;\Omega}) \end{align*}
Is my proof correct ? Thanks for your answers !