updated elliptic pdes and removed evolution pdes

Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex

@@ -223,5 +223,26 @@
   \begin{definition}
     If the weak solution $u_f$ of the problem $\mathcal{D}_f$ is in $H^1_0(\Omega)\cap W^{2,p}(\Omega)$ for some $p\in [1,\infty)$, then we say that $u_f$ is called a \emph{strong solution} of $\mathcal{D}_f$. If $u_f\in \mathcal{C}^2(\Omega)\cap H^1_0(\Omega)$, then we say that $u_f$ is a \emph{classical solution} of $\mathcal{D}_f$.
   \end{definition}
+  \begin{theorem}
+    Let $1<p<\infty$ and $\Omega\subset\RR^d$ be open and bounded with $\mathcal{C}^{m+1}$ boundary, $m\geq 1$. Let $a_{ij}\in \mathcal{C}^m(\overline{\Omega})$, $b_j,c\in \mathcal{C}^{m-1}(\overline{\Omega})$ and $Lu=-\sum_{i,j=1}^d \partial_i(a_{ij}\partial_j u)+\sum_{j=1}^d b_j\partial_j u+cu$ be an elliptic operator. Then, for any $f\in W^{m-1,p}(\Omega)$, if $u\in H^1_0(\Omega)$ is a weak solution of $\mathcal{D}_f$, then $u\in W^{m+1,p}(\Omega)$ and:
+    $$
+      \norm{u}_{W^{m+1,p}(\Omega)}\leq C\left(\norm{f}_{W^{m-1,p}(\Omega)}+\norm{u}_{L^2(\Omega)}\right)
+    $$
+    If in addition the weak solution of $\mathcal{D}_0$ is $u=0$, then $L:W^{m+1,p}(\Omega)\cap W_0^{1,p}(\Omega)\to W^{m-1,p}(\Omega)$ is an isomorphism, where $W_0^{1,p}(\Omega)$ is the closure of $C_0^\infty(\Omega)$ in $W^{1,p}(\Omega)$.
+  \end{theorem}
+  \subsection{Regularity in \texorpdfstring{$\mathcal{C}^{k,\alpha}$}{Ckalpha} for divergence-form elliptic PDEs}
+  In this section we will not use the usual Hölder norm
+  $$
+    \norm{u}_{\mathcal{C}^{k,\alpha}(\overline\Omega)}=\sup_{\substack{x\ne y\\ \abs{\beta}=k}}\frac{\abs{\partial^\beta u(x)-\partial^\beta u(y)}}{\abs{x-y}^\alpha}
+  $$
+  but the following one:
+  $$
+    \norm{u}_{\mathcal{C}^{k,\alpha}(\overline\Omega)}=\sup_{\substack{x\in \overline\Omega\\ \abs{\beta}\leq k}}\abs{\partial^\beta u(x)}+\sup_{\substack{x\ne y\\ \abs{\beta}=k}}\frac{\abs{\partial^\beta u(x)-\partial^\beta u(y)}}{\abs{x-y}^\alpha}
+  $$
+
+  \begin{remark}
+    Recall that $(\mathcal{C}^{k,\alpha}(\overline\Omega),\norm{\cdot}_{\mathcal{C}^{k,\alpha}(\overline\Omega)})$ is a Banach space and that if $0<\alpha_1\leq\alpha_2<1$, then $
+      \mathcal{C}^{k,\alpha_2}(\overline{\Omega})\subseteq \mathcal{C}^{k,\alpha_1}(\overline{\Omega})$
+  \end{remark}
 \end{multicols}
 \end{document}

index.html

@@ -73,8 +73,8 @@ <h2 class="special-color">Mathematics</h2>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Advanced_dynamical_systems.pdf';" target="_top">Advanced dynamic systems</button></li>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Advanced_probability.pdf';" target="_top">Advanced probability</button></li>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Advanced_topics_in_functional_analysis_and_PDEs.pdf';" target="_top">Advanced topics in functional analysis and PDEs</button></li>
-          <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Introduction_to_evolution_PDEs.pdf';" target="_top">Introduction to evolution PDEs</button></li>
-          <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Introduction_to_nonlinear_PDEs.pdf';" target="_top">Introduction to non linear PDEs</button></li>
+          <!-- <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Introduction_to_evolution_PDEs.pdf';" target="_top">Introduction to evolution PDEs</button></li> -->
+          <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Introduction_to_nonlinear_elliptic_PDEs.pdf';" target="_top">Introduction to nonlinear elliptic PDEs</button></li>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Montecarlo_methods.pdf';" target="_top">Montecarlo methods</button></li>
           <li><button class="button" onclick="window.location.href='https://github.com/victorballester7/Complete-summaries/releases/latest/download/Stochastic_calculus.pdf';" target="_top">Stochastic calculus</button></li>
         </ul>

main_math.tex

@@ -110,9 +110,6 @@ \chapter{Fifth year}
 \subfile{Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex}
 \cleardoublepage
 
-% \subfile{Mathematics/5th/Introduction_to_evolution_PDEs/Introduction_to_evolution_PDEs.tex}
-% \cleardoublepage
-
 \subfile{Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex}
 \cleardoublepage