improve little pdes

Mathematics/5th/Introduction_to_nonlinear_elliptic_PDEs/Introduction_to_nonlinear_elliptic_PDEs.tex

@@ -238,7 +238,7 @@
     $$
       f= (L_{\mu_0}-\mu_0){L_{\mu_0}}^{-1}w=w-\mu_0 {L_{\mu_0}}^{-1}w=(\id-K)w
     $$
-    with $K=\mu_0 {L_{\mu_0}}^{-1}$. We claim that $K:L^2(\Omega)\to L^2(\Omega)$ is compact. Note that $K=\iota_{H_0^1\hookrightarrow L^2}\circ {L_{\mu_0}}^{-1}\circ \iota_{L^2\hookrightarrow H^{-1}}$, so since ${L_{\mu_0}}^{-1}$ and $\iota_{L^2\hookrightarrow H^{-1}}$ are bounded and we have a compact embedding $H_0^1\hookrightarrow L^2$, we have that $K$ is compact. Finally, one can check that:
+    with $K=\mu_0 {L_{\mu_0}}^{-1}$. We claim that $K:L^2(\Omega)\to L^2(\Omega)$ is compact. Note that $K=\iota_{H_0^1\hookrightarrow L^2}\circ {L_{\mu_0}}^{-1}\circ \iota_{L^2\hookrightarrow H^{-1}}$, so since ${L_{\mu_0}}^{-1}$ and $\iota_{L^2\hookrightarrow H^{-1}}$ are bounded, and we have a compact embedding $H_0^1\hookrightarrow L^2$, we have that $K$ is compact. Finally, one can check that:
     $$
       \id-K^*={(\id-K)}^*={\left(L_0{L_{\mu_0}}^{-1}\right)}^*={({L_{\mu_0}}^*)}^{-1}{L_0}^{*}
     $$