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 974725a..4f3d529 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 @@ -182,7 +182,7 @@ \grad(u^+)\almoste{=}\begin{cases} \grad u & \text{if }u>0 \\ 0 & \text{if }u\leq 0 - \end{cases}\quad + \end{cases}\;\; \grad(u^-)\almoste{=}\begin{cases} -\grad u & \text{if }u<0 \\ 0 & \text{if }u\geq 0 @@ -256,6 +256,10 @@ $$ \norm{u}_{\mathcal{C}^{2,\alpha}(\overline{\Omega})}\leq C\left(\norm{f}_{\mathcal{C}^{0,\alpha}(\overline{\Omega})}+\norm{u}_{\mathcal{C}^{1,\alpha}(\overline{\Omega})}\right) $$ + Moreover we have: + $$ + \norm{u}_{\mathcal{C}^{2,\alpha}(\overline{\Omega})}\leq \tilde{C}\left(\norm{f}_{\mathcal{C}^{0,\alpha}(\overline{\Omega})}+\norm{u}_{\mathcal{C}^{0}(\overline{\Omega})}\right) + $$ \end{theorem} \begin{corollary} Let $\Omega\subset \RR^d$ be open and bounded with $\Fr{\Omega}\in\mathcal{C}^{k+2,\alpha}$ for some $0<\alpha<1$ and $k\geq 0$. In the elliptic operator $L$ assume that $a_{ij},b_j,c\in\mathcal{C}^{k,\alpha}(\overline{\Omega})$. Then, $\exists c>0$ such that if $u\in\mathcal{C}^{k+2}(\Omega)\cap \mathcal{C}^k(\overline{\Omega})$ solves $Lu=f$, with $f\in\mathcal{C}^{k,\alpha}(\overline{\Omega})$, then $u\in \mathcal{C}^{k+2,\alpha}(\overline{\Omega})$ and: @@ -309,5 +313,33 @@ $$ with $f,a_{ij},b_j,c\in\mathcal{C}^{0,\alpha}( \overline{\Omega})$ and $h\in\mathcal{C}^{0,\alpha}(\partial\Omega)$. Then, there exists a solution to this problem in $\mathcal{C}^{2,\alpha}(\overline{\Omega})$. \end{theorem} + \subsection{Existence theorems for nonlinear elliptic PDEs by fixed point methods} + In this section we will mostly consider almost linear elliptic PDEs of the form: + \begin{equation}\label{INLEPDE:AlmostLinear} + \begin{cases} + Lu=f(x,u) \\ + u|_{\partial\Omega}=0 + \end{cases} + \end{equation} + with $L$ either $-\sum_{i,j=1}^d\partial_i(a_{ij}\partial_j)+\sum_{j=1}^db_j\partial_j$ or $-\sum_{i,j=1}^d a_{ij} \partial_{ij}^2+\sum_{j=1}^db_j\partial_j$, and $f:\Omega\times \RR\to \RR$. + \subsubsection{Method of subsoltions and supersolutions} + \begin{theorem} + Suppose that an operator $L$ is uniformly elliptic on an open bounded set $\Omega\subset\RR^d$ with $\Fr{\Omega}\in \mathcal{C}^2$, with $c=0$ and either in divergence form (with $a_{ij}\in\mathcal{C}^1$) or non-divergence form (with $a_{ij},b_j\in\mathcal{C}^{0,\alpha}$). Suppose that $f\in\mathcal{C}^1(\overline{\Omega}\times \RR)$ and assume that the problem of \mcref{INLEPDE:AlmostLinear} has a bounded subsolution $\underline{u}$ and a bounded supersolution $\overline{u}$ such that $\underline{u}\leq \overline{u}$. Then, there exists a solution $u$ to \mcref{INLEPDE:AlmostLinear} such that $\underline{u}\leq u\leq \overline{u}$, which is in $H_0^1(\Omega)\cap H_0^2(\Omega)$ if $L$ is in divergence form and in $\mathcal{C}^{2,\alpha}(\overline{\Omega})$ if $L$ is in non-divergence form. + \end{theorem} + \subsubsection{Topological fixed point theorems} + \begin{theorem}[Brower fixed point] + Let $C\subset \RR^n$ be a closed convex bounded set and $f:C\to C$ be a continuous function. Then, $f$ has at least a fixed point. + \end{theorem} + \begin{theorem}[Schauder fixed point] + Let $C$ be a convex set in a Banach space $(E,\norm{\cdot})$ and $f:C\to C$ be a continuous function. Assume one of the following two assumptions: + \begin{itemize} + \item $C$ is compact for $\norm{\cdot}$. + \item $C$ is closed and bounded and $f$ is compact. + \end{itemize} + Then, $f$ has at least a fixed point. + \end{theorem} + \begin{theorem}[Schaefer fixed point] + Let $(E, \norm{\cdot})$ be Banach and $f:E\to E$ be a continuous and compact. Suppose that $\exists M>0$ such that $\forall (\lambda,u)\in [0,1]\times E$ with $u=\lambda f(u)$ we have $\norm{u}