I am stuck at establishing regularity for elliptic equations on the one dimensional domain $\Omega=[0,1]$. The problem is $Lu=f$, in $\Omega$, and $u(0)=0, u_x(1)=0.$ In the page 317, Evans' book, 1997, there is only the regularity for the Dirichlet boundary condition (The Boundary $H^2$- regularity). How can I get the same estimate for my problem? Thank you!
$\begingroup$
$\endgroup$
6
-
$\begingroup$ There are a standard results in ODE's covering your problem and you should ask it on math.stackexchange.com However, you should explain what is $L$ since the answer depends on what you assume about $L$. $\endgroup$– Piotr HajlaszJun 22, 2018 at 13:40
-
$\begingroup$ Thank you, Piotr, $L$ in my case is the same in Evans' book. It is elliptic with $Lu=-a(x)u_{xx} + b(x)u_x+c(x)u$. $\endgroup$– thnguyenJun 22, 2018 at 13:46
-
$\begingroup$ However, I am expecting the results for weak solution, I don't know whether ODE's results covers it. The estimate I want to obtain is the same in Evans' book: $\Vert u \Vert_{H^2(\Omega)} \leq C\left( \Vert f\Vert_{L^2(\Omega)}+\Vert u\Vert_{L^2(\Omega)}\right)$ $\endgroup$– thnguyenJun 22, 2018 at 13:55
-
$\begingroup$ What do you assume about $a$, $b$ and $c$. If you ask a question you need to be precise. $\endgroup$– Piotr HajlaszJun 22, 2018 at 15:31
-
$\begingroup$ Thank you Piotr! Let's assume them first like in the Theorem 4, p. 317, Evans: $a\in C^1(\bar{\Omega}), b, c\in L^\infty(\Omega)$ and $f\in L^2(\Omega)$. $\endgroup$– thnguyenJun 22, 2018 at 15:55
|
Show 1 more comment