From d1f160fa349863b26bfc09ae45b4bf2bab86316d Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Sun, 31 Dec 2023 15:08:05 +0100 Subject: [PATCH] typos --- .../Introduction_to_nonlinear_elliptic_PDEs.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 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 80e511d..8a32197 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 @@ -1069,9 +1069,9 @@ \begin{proof} From the superquadradicity condition, for $t>0$, the function $\abs{t}^{-p}F(x,t)$ is nondecreasing (the derivative is nonnegative). So, for $0\leq t\leq 1$ we have $F(x,t)\leq \abs{t}^p F(x,1)$. Similarly, for $-1\leq t\leq 0$ we have $F(x,t)\leq \abs{t}^p F(x,-1)$. Using the upper estimate we get, for $\abs{t}\geq 1$, $\abs{F(x,t)}\leq \overline{\overline{C}} \abs{t}^{p_1}$ and so $\abs{F(x,t)}\leq C'(\abs{t}^p+\abs{t}^{p_1})$ $\forall t$ (the positivity of $F(x,1)$ and $F(x,-1)$ follows from the third hypothesis on $f$). So: - $$ - \int_\Omega \abs{F(x,u)}\leq C'\left(\norm{u}_{ - $$ + % $$ + % \int_\Omega \abs{F(x,u)}\leq C'\left(\norm{u}_{ + % $$ \end{proof} \end{multicols} \end{document} \ No newline at end of file