From 233bdbff438e6b54534a0817924127657a951ae3 Mon Sep 17 00:00:00 2001 From: George Stamatiou Date: Fri, 12 Jan 2024 14:57:59 +0100 Subject: [PATCH] Corrected regularity condition on regularity up to the boundary for uniformly elliptic PDEs --- .../Introduction_to_nonlinear_elliptic_PDEs.tex | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex b/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex index ba3d635..f3c0667 100644 --- a/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex +++ b/Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex @@ -352,7 +352,7 @@ $$ \end{theorem} \begin{corollary} - Assume that $\Fr{\Omega}$ is $\mathcal{C}^{m}$, $m\in\NN$, and that $a_{ij}\in \mathcal{C}^{m+1}(\overline{\Omega})$, $b_j,c\in \mathcal{C}^{m}(\overline{\Omega})$. Let $f\in H^{m}(\Omega)$ and $u\in H^1_0(\Omega)$ be a weak solution of $\mathcal{D}_f$. Then, $u\in H^{m+2}(\Omega)$ and: + Assume that $\Fr{\Omega}$ is $\mathcal{C}^{m+2}$, $m\in\NN$, and that $a_{ij}\in \mathcal{C}^{m+1}(\overline{\Omega})$, $b_j,c\in \mathcal{C}^{m}(\overline{\Omega})$. Let $f\in H^{m}(\Omega)$ and $u\in H^1_0(\Omega)$ be a weak solution of $\mathcal{D}_f$. Then, $u\in H^{m+2}(\Omega)$ and: $$ \norm{u}_{H^{m+2}(\Omega)}\leq C\left(\norm{f}_{H^{m}(\Omega)}+\norm{u}_{L^2(\Omega)}\right) $$