From 0d28ad20180f8918addee85eb9a9ad59d6d92389 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?V=C3=ADctor?= <victor.ballester.ribo@gmail.com>
Date: Sun, 19 Nov 2023 19:55:02 +0100
Subject: [PATCH] updated typos

---
 .../Introduction_to_nonlinear_elliptic_PDEs.tex   | 15 ++++++---------
 .../Numerical_methods_for_PDEs.tex                |  2 +-
 2 files changed, 7 insertions(+), 10 deletions(-)

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 0cf735f..a345b0b 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
@@ -314,10 +314,7 @@
     $$
   \end{corollary}
   \begin{corollary}
-    Assume $a_{ij},b_j,c,f\in\mathcal{C}^\infty(\Omega)$. Let $u\in H^1(\Omega)$ be a weak solution of $Lu=f$. Then, $u\in \mathcal{C}^\infty(\Omega)$ and $\forall m\in \NN$:
-    $$
-      \norm{u}_{H^{m}(\Omega)}\leq C\left(\norm{f}_{H^{m}(\Omega)}+\norm{u}_{L^2(\Omega)}\right)
-    $$
+    Assume $a_{ij},b_j,c,f\in\mathcal{C}^\infty(\Omega)$. Let $u\in H^1(\Omega)$ be a weak solution of $Lu=f$. Then, $u\in \mathcal{C}^\infty(\Omega)$.
   \end{corollary}
   \begin{theorem}[Regularity up to the boundary]
     Assume that $\Fr{\Omega}$ is $\mathcal{C}^2$ and that $a_{ij}\in \mathcal{C}^1(\overline{\Omega})$, $b_j,c\in L^\infty(\Omega)$. Let $f\in L^2(\Omega)$ and $u\in H^1_0(\Omega)$ be a weak solution of $\mathcal{D}_f$. Then, $u\in H^2(\Omega)$ and:
@@ -342,12 +339,12 @@
     Let $\Omega\subseteq\RR^d$ open and $u\in H^1(\Omega)$. Then:
     $$
       u^{+}:=\begin{cases}
-        u & \text{if }u\geq 0 \\
-        0 & \text{if }u<0
+        u & \text{if }u> 0    \\
+        0 & \text{if }u\leq 0
       \end{cases}\qquad
       u^{-}:=\begin{cases}
-        -u & \text{if }u\leq 0 \\
-        0  & \text{if }u>0
+        -u & \text{if }u< 0    \\
+        0  & \text{if }u\leq 0
       \end{cases}
     $$
     are also in $H^1(\Omega)$ and:
@@ -366,7 +363,7 @@
     Let $\Omega\subseteq\RR^d$ open and $u\in H^1(\Omega)$. Then, $\abs{u}\in H^1(\Omega)$ and $\grad{\abs{u}}=\sign\grad{u}$.
   \end{corollary}
   \begin{lemma}
-    Let $(u_n)\in H^1(\Omega)$ be such that $u_n\overset{H^1(\Omega)}{\longrightarrow} u$. Then, $u_n^\pm\overset{H^1(\Omega)}{\longrightarrow} u^\pm$.
+    Let $(u_n)\in H^1(\Omega)$ be such that $u_n\overset{H^1(\Omega)}{\longrightarrow} u$. Then, ${u_n}^\pm\overset{H^1(\Omega)}{\longrightarrow} u^\pm$.
   \end{lemma}
   \begin{corollary}
     Let $u\in H^1(\Omega)$. Then, $\Tr_{\partial\Omega}(u^\pm)={(\Tr_{\partial\Omega}u)}^\pm$.
diff --git a/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex b/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex
index 2e796a3..989c72a 100644
--- a/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex
+++ b/Mathematics/5th/Numerical_methods_for_PDEs/Numerical_methods_for_PDEs.tex
@@ -285,7 +285,7 @@
     $$
   \end{proposition}
   \begin{remark}
-    Note that ${(P_Nu)}'=P_Nu'$, but ${(I_Nu)}'\neq I_Nu'$. What we do in generall is to pass to the Fourier space, differentiate and then come back to the physical space.
+    Note that ${(P_Nu)}'=P_Nu'$, but ${(I_Nu)}'\neq I_Nu'$. What we do in general is to pass to the Fourier space, differentiate and then come back to the physical space.
   \end{remark}
   \subsubsection{Non-periodic problems}
   \begin{remark}