still more changes

Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex

@@ -231,7 +231,7 @@
     \begin{equation*}
       {\langle \grad v,\grad w\rangle}_2+{\langle \vf{b}\cdot \grad v,w\rangle}_2=0\quad \forall w\in H_0^1(\Omega)
     \end{equation*}
-    has a finite dimensional solution space $W_0$, as well as the space $V_0$ of solutions of $\mathcal{D}_0$, and $\dim W_0=\dim V_0$. Moreover, if $f\in L^2(\Omega)$, $\mathcal{D}_f$ is solvable if and only if $\langle f,v\rangle=0$ for all $v\in W_0$.
+    has a finite dimensional solution space $W_0$, the space $V_0$ of solutions of $\mathcal{D}_0$ has also finite dimension and $\dim W_0=\dim V_0$. Moreover, if $f\in L^2(\Omega)$, $\mathcal{D}_f$ is solvable if and only if $\langle f,v\rangle=0$ for all $v\in W_0$.
   \end{proposition}
   \begin{proof}
     We saw in \mcref{INEPDE:Lmu} that for $\mu\geq \mu_0>0$, $L_\mu$ is an isomorphism. Now we want to solve $L_0u=f$. Consider the change of variables $u={L_{\mu_0}}^{-1}w$, with $w=L_{\mu_0}u\in H^{-1}(\Omega)$. Thus, the equation becomes:
@@ -242,7 +242,7 @@
     $$
       \id-K^*={(\id-K)}^*={\left(L_0{L_{\mu_0}}^{-1}\right)}^*={({L_{\mu_0}}^*)}^{-1}{L_0}^{*}
     $$
-    By \mcref{INEPDE:fredholm}, we have that $(\id-K)w=f$ has a solution if and only if $(\id-K^*)h=0\implies \langle f,h\rangle_{L^2} =0$ for all $h\in L^2$. But ${L_{\mu_0}}^*$ is an isomorphism, so $(\id-K^*)h=0\iff {L_0}^*h=0$.
+    By \mnameref{INEPDE:fredholm}, we have that $(\id-K)w=f$ has a solution if and only if $(\id-K^*)h=0\implies \langle f,h\rangle_{L^2} =0$ for all $h\in L^2$. But ${L_{\mu_0}}^*$ is an isomorphism, so $(\id-K^*)h=0\iff {L_0}^*h=0$.
   \end{proof}
   \begin{definition}
     We define the following problem: