updated pdes and stochastic calculus

Mathematics/3rd/Probability/Probability.tex

@@ -730,7 +730,7 @@
     Check the proof of \mnameref{RFA:monotone}.
   \end{sproof}
   \begin{theorem}[Dominated convergence theorem]\label{P:dominated}
-    Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(X_n)$ be sequence of random variables such that $\displaystyle\lim_{n\to\infty}X_n\overset{\text{a.s.}}{=}X$, for some random variable $X$. Suppose that there exists an integrable random variable $Y$ such that $$|X_n|\leq Y\quad\forall n\geq 1$$ Then: $$\lim_{n\to\infty} \Exp(X_n)=\Exp(X)$$
+    Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(X_n)$ be sequence of random variables such that $\displaystyle\lim_{n\to\infty}X_n\overset{\text{a.s.}}{=}X$, for some random variable $X$. Suppose that there exists an integrable random variable $Y$ such that $$\abs{X}\leq Y\quad\forall n\geq 1$$ Then: $$\lim_{n\to\infty} \Exp(X_n)=\Exp(X)$$
   \end{theorem}
   \begin{sproof}
     Check the proof of \mnameref{RFA:dominated}.
@@ -1162,21 +1162,21 @@
     Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(X_n)$ be a sequence of random variables. We define the sequence of partial sums $(S_n)$ as: $$S_n:=\sum_{i=1}^nX_i$$
   \end{definition}
   \subsubsection{Weak laws}
-  \begin{theorem}[Weak law]
+  \begin{theorem}[Weak law of large numbers]\label{P:weaklaw}
     Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(X_n)$ be a sequence of \iid random variables with finite 2nd moment. Then: $$\frac{S_n}{n}\overset{\Prob}{\longrightarrow}\Exp(X_1)\quad\text{and}\quad\frac{S_n}{n}\overset{L^2}{\longrightarrow}\Exp(X_1)$$
   \end{theorem}
-  \begin{theorem}[Weak law]
+  \begin{theorem}[Weak law of large numbers]
     Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(X_n)$ be a sequence of pairwise uncorrelated random variables with finite 2nd moment. Suppose that: $$\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^n\Exp(X_i)=\mu<\infty\;\,\text{and}\;\,\lim_{n\to\infty}\frac{1}{n^2}\sum_{i=1}^n\Var(X_i)=0$$ Then: $$\frac{S_n}{n}\overset{\Prob}{\longrightarrow}\mu\quad\text{and}\quad\frac{S_n}{n}\overset{L^2}{\longrightarrow}\mu$$
   \end{theorem}
   \subsubsection{Strong laws}
-  \begin{theorem}[Kolmogorov's strong law]\label{P:stronglawKolmo}
+  \begin{theorem}[Kolmogorov's strong law of large numbers]\label{P:stronglawKolmo}
     Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(X_n)$ be a sequence of \iid random variables.
     \begin{enumerate}
       \item If $\Exp(X_1)<\infty$, then: $$\frac{S_n}{n}\overset{\text{a.s.}}{\longrightarrow}\Exp(X_1)$$
       \item If $\Exp(X_1)=\infty$, then: $$\limsup_{n\to\infty}\frac{|S_n|}{n}\overset{\text{a.s.}}{=}+\infty$$
     \end{enumerate}
   \end{theorem}
-  \begin{theorem}[Strong law]
+  \begin{theorem}[Strong law of large numbers]
     Let $(\Omega,\mathcal{A},\Prob)$ be a probability space and $(X_n)$ be a sequence of \iid random variables such that $\Exp({X_1}^4)<\infty$. Then: $$\frac{S_n}{n}\overset{\text{a.s.}}{\longrightarrow}\Exp(X_1)$$
   \end{theorem}
   \begin{corollary}

Mathematics/5th/Advanced_topics_in_functional_analysis_and_PDEs/Advanced_topics_in_functional_analysis_and_PDEs.tex

@@ -247,13 +247,13 @@
     From now on, we will denote $\RR_{\pm}^d:=\RR^{d-1}\times\RR_{\pm}$ and $\RR_0^d:=\RR^{d-1}\times\{0\}$.
   \end{remark}
   \begin{theorem}
-    For all $m\in\NN$ and all $1\leq p<\infty$, $\mathcal{C}^\infty(\overline{\RR_+^d})$ is dense in $W^{m,p}(\RR_+^d)$.
+    For all $m\in\NN$ and all $1\leq p<\infty$, $\mathcal{C}^\infty(\overline{\RR_{\geq 0}^d})$ is dense in $W^{m,p}(\RR_{\geq 0}^d)$.
   \end{theorem}
   \begin{proof}
     Let $$
       \tau_h(u)(x_1,\ldots, x_d):=u(x_1,\ldots, x_{d-1},x_d+h)
     $$
-    be the translation operator and set $u_\varepsilon:=\tau_{\varepsilon}(u)*\phi_\varepsilon$, where $\varepsilon>0$ and $\phi_\varepsilon$ is an approximation of identity. Then, $u_\varepsilon\in \mathcal{C}^\infty(\overline{\RR_+^d})$ by the properties of the convolution. Moreover:
+    be the translation operator and set $u_\varepsilon:=\tau_{\varepsilon}(u)*\phi_\varepsilon$, where $\varepsilon>0$ and $\phi_\varepsilon$ is an approximation of identity. Then, $u_\varepsilon\in \mathcal{C}^\infty(\overline{\RR_{\geq 0}^d})$ by the properties of the convolution. Moreover:
     \begin{multline*}
       \norm{\partial^\alpha u_\varepsilon-\partial^\alpha u}_p \leq \norm{\partial^\alpha u_\varepsilon-\partial^\alpha (\tau_{\varepsilon}u)}_p+\norm{\partial^\alpha (\tau_{\varepsilon}u)-\partial^\alpha u}_p                         \\
       \leq \norm{(\partial^\alpha \tau_\varepsilon u)*\phi_\varepsilon-\partial^\alpha (\tau_{\varepsilon}u)}_p+\norm{\tau_{\varepsilon}(\partial^\alpha u)-\partial^\alpha u}_p
@@ -264,24 +264,24 @@
     The same proof shows that $\mathcal{C}^\infty(\overline{\Omega})$ is dense in $W^{m,p}(\Omega)$, if $\Omega$ is bounded with $\Fr{\Omega}$ of class $\mathcal{C}^1$. This time, one needs to locally translate u along the normal direction.
   \end{remark}
   \begin{theorem}
-    For all $m\in\NN$ and all $1\leq p<\infty$, there is an extension operator $E:W^{m,p}(\RR_+^d)\to W^{m,p}(\RR^d)$.
+    For all $m\in\NN$ and all $1\leq p<\infty$, there is an extension operator $E:W^{m,p}(\RR_{\geq 0}^d)\to W^{m,p}(\RR^d)$.
   \end{theorem}
   \begin{proof}
-    We only do the proof for $d=1$ and $m=1$ to highlight the main ideas. Let $u\in W^{1,p}(\RR_+)$. We define the \emph{first order reflection}:
+    We only do the proof for $d=1$ and $m=1$ to highlight the main ideas. Let $u\in W^{1,p}(\RR_{\geq 0})$. We define the \emph{first order reflection}:
     $$
       \bar{u}:=\begin{cases}
         u(x)             & \text{if }x\geq 0 \\
         -3u(-x)+4u(-x/2) & \text{if }x<0
       \end{cases}
     $$
-    By density, it is enough to prove the result for $u\in \mathcal{C}^1(\RR_+)$. An easy check shows that $\bar{u}\in \mathcal{C}^1(\RR)$. Moreover, we have:
+    By density, it is enough to prove the result for $u\in \mathcal{C}^1(\RR_{\geq 0})$. An easy check shows that $\bar{u}\in \mathcal{C}^1(\RR)$. Moreover, we have:
     \begin{align*}
-      {\norm{\bar{u}}_{W^{1,p}(\RR)}}^p & =\int_{\RR}{\abs{\bar{u}}^p}+{\abs{\bar{u}'}^p}             \\
+      {\norm{\bar{u}}_{W^{1,p}(\RR)}}^p & =\int_{\RR}{\abs{\bar{u}}^p}+{\abs{\bar{u}'}^p}                                                 \\
       \begin{split}
-        &=\!\int_{\RR_+}\!{\abs{u}^p}\!+\!\!{\abs{u'}^p}\!+\!\int_{\RR_-}\![{\abs{-3u(-x)+4u(-x/2)}^p}+\\
+        &=\!\int_{\RR_{\geq 0}}\!{\abs{u}^p}\!+\!\!{\abs{u'}^p}\!+\!\int_{\RR_{\leq 0}}\![{\abs{-3u(-x)+4u(-x/2)}^p}+\\
         &\hspace{2.75cm}+{\abs{3u'(-x)-2u'(-x/2)}^p}]
       \end{split} \\
-                                        & \leq C{\norm{u}_{W^{1,p}(\RR_+)}}^p
+                                        & \leq C{\norm{u}_{W^{1,p}(\RR_{\geq 0})}}^p
     \end{align*}
     for some constant $C>0$. Thus, $E$ is a bounded extension operator.
   \end{proof}
@@ -300,7 +300,7 @@
     The proof for higher derivatives $m \geq 1$ needs to add more terms in order to make the junction smooth enough.
   \end{remark}
   \begin{definition}
-    We say that a domain $\Omega\subseteq \RR^d$ has boundary of class $\mathcal{C}^k$ if $\forall x\in \Fr{\Omega}$ there is a neighborhood $\varepsilon,\delta>0$ and a $\mathcal{C}^k$-diffeomorphism $\phi:B(x,\varepsilon)\to B(0,\delta)$ so that $\phi(x)=0$ and $\phi(B(x,\varepsilon)\cap \Omega)=B(0,\delta)\cap \RR_+^d$. Note that in particular this implies that $\phi(\Fr{\Omega}\cap B(x,\varepsilon))=B(0,\delta)\cap \RR_0^d$.
+    We say that a domain $\Omega\subseteq \RR^d$ has boundary of class $\mathcal{C}^k$ if $\forall x\in \Fr{\Omega}$ there is a neighborhood $\varepsilon,\delta>0$ and a $\mathcal{C}^k$-diffeomorphism $\phi:B(x,\varepsilon)\to B(0,\delta)$ so that $\phi(x)=0$ and $\phi(B(x,\varepsilon)\cap \Omega)=B(0,\delta)\cap \RR_{\geq 0}^d$. Note that in particular this implies that $\phi(\Fr{\Omega}\cap B(x,\varepsilon))=B(0,\delta)\cap \RR_0^d$.
   \end{definition}
   \begin{theorem}
     Let $\Omega\subseteq \RR^d$ be a bounded domain with $\mathcal{C}^k$ boundary. Then, $\forall m\leq k$ and all $1\leq p<\infty$, there is an extension operator $E:W^{m,p}(\Omega)\to W^{m,p}(\RR^d)$.
@@ -320,16 +320,16 @@
   \end{theorem}
   \subsubsection{Trace operators}
   \begin{theorem}
-    Let $1\leq p<\infty$ and $u\in W^{1,p}(\RR_+^d)$. Then, the function $u|_{\RR_0^d}:\RR^{d-1}\to\CC$ belongs to $L^p(\RR^{d-1})$.
+    Let $1\leq p<\infty$ and $u\in W^{1,p}(\RR_{\geq 0}^d)$. Then, the function $u|_{\RR_0^d}:\RR^{d-1}\to\CC$ belongs to $L^p(\RR^{d-1})$.
   \end{theorem}
   \begin{definition}
     We define the \emph{trace operator} as the map:
     \begin{align*}
-      \function{T}{W^{1,p}(\RR_+^d)}{L^p(\RR^{d-1})}{u}{u|_{\RR_0^d}}
+      \function{T}{W^{1,p}(\RR_{\geq 0}^d)}{L^p(\RR^{d-1})}{u}{u|_{\RR_0^d}}
     \end{align*}
   \end{definition}
   \begin{theorem}
-    Let $1\leq p<\infty$ and $u\in W^{1,p}(\RR_+^d)$. Then, $Tu=0$ if and only if $u\in W_0^{1,p}(\RR_+^d)$.
+    Let $1\leq p<\infty$ and $u\in W^{1,p}(\RR_{\geq 0}^d)$. Then, $Tu=0$ if and only if $u\in W_0^{1,p}(\RR_{\geq 0}^d)$.
   \end{theorem}
   \begin{lemma}
     Let $\Omega\subseteq \RR^d$ be an open set and $u\in W^{1,p}(\Omega)$ with $1\leq p\leq \infty$. Then, $\norm{\grad \abs{u}}\almoste{\leq}\norm{\grad u}$.

Mathematics/5th/Introduction_to_non_linear_elliptic_PDEs/Introduction_to_non_linear_elliptic_PDEs.tex

@@ -3,6 +3,49 @@
 \begin{document}
 \changecolor{INLEPDE}
 \begin{multicols}{2}[\section{Introduction to non linear elliptic PDEs}]
-
+  \subsection{Introduction}
+  \begin{definition}
+    Let $a_{ij}$, $b_j$, $c$, $f$ be known functions on $\Omega\subseteq \RR^d$. Usually we will denote $\vf{A}=(a_{ij})$ and $\vf{b}=(b_j)$ A \emph{linear second-order PDE} is an equation of the form:
+    \begin{equation*}
+      -\sum_{i,j=1}^da_{ij}(x){\partial_{ij}}^2u(x)+\sum_{j=1}^db_j(x)\partial_ju(x)+c(x)u(x)=f(x)
+    \end{equation*}
+    where $u:\Omega\to \RR$ is the unknown function. This form is called \emph{non-divergence form}. If we write the equation in the form:
+    \begin{multline*}
+      -\sum_{i=1}^d\pdv{}{x_i}\left(\sum_{j=1}^da_{ij}(x)\partial_ju(x)\right)+\sum_{j=1}^db_j(x)\partial_ju(x)+\\+c(x)u(x)=f(x)
+    \end{multline*}
+    then we say that the equation is in \emph{divergence form}. Together with the PDE we usually impose boundary conditions on $\partial\Omega$. The \emph{Dirichlet boundary condition} is:
+    $$
+      u|_{\partial\Omega}=g
+    $$
+    and it is called \emph{homogeneous} if $g=0$. The \emph{Neumann boundary condition} is:
+    $$
+      \langle \vf{n},\vf{A} \nabla u\rangle|_{\partial\Omega}=g
+    $$
+    where we have assumed that the boundary of $\Omega$ is smooth enough to define the normal vector $\vf{n}$. The condition is called \emph{homogeneous} if $g=0$. Note that if $\vf{A}=\vf{I}_d$, then the Neumann boundary condition is just $\partial_{\vf{n}} u=g$.
+  \end{definition}
+  \begin{definition}
+    Let $a_{ij},b_j,c$ be known functions on $\Omega\subseteq \RR^d$. We say that the operator $$L=-\sum_{i,j=1}^da_{ij}{\partial_{ij}}^2 + \sum_{j=1}^d b_j\partial_j+c$$ is \emph{uniformly elliptic} if there exists $\theta>0$ such that for all $x\in \Omega$ and all $p\in \RR^d$ we have:
+    \begin{equation}
+      Q_x(p)=\sum_{i,j=1}^da_{ij}(x)p_ip_j\geq \theta \sum_{i=1}^{d} {p_i}^2
+    \end{equation}
+  \end{definition}
+  \begin{remark}
+    Geometrically speaking, this implies that the sets
+    $$
+      \xi_{x,h}=\{ p\in \RR^d: Q_x(p)=h\}
+    $$
+    are ellipsoids.
+  \end{remark}
+  \subsection{Hilbert space methods for divergence form linear PDEs}
+  In this section, we will assume that $\Omega\subset\RR^d$ is an open, bounded subset, $a_{ij}=a_{ji}$ and $a_{ij},b_j,c\in L^\infty(\Omega)$.
+  \subsubsection{Fredholm alternative}
+  \begin{theorem}[Abstract Fredholm alternative]
+    Let $H$ be Hilbert and $K:H\to H$ be a compact linear operator. Then:
+    \begin{enumerate}
+      \item $\ker(\id-K)$ and $\ker(\id-K^*)$ are both finite dimensional and they have the same dimension.
+      \item $\im(\id-K)={(\ker(\id-K^*))}^\perp$. In particular, $\im(\id-K)$ is closed.
+      \item Either $\ker(\id-K)\ne\{0\}$ or $\id -K$ is and isomorphism.
+    \end{enumerate}
+  \end{theorem}
 \end{multicols}
 \end{document}