1
$\begingroup$

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!

$\endgroup$
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 $L$. $\endgroup$ Jun 22, 2018 at 13:40
  • $\begingroup$ Thank you, Piotr, $L$ in my case is the same in Evans' book. It is elliptic with $Lu=-a(x)u_{xx} + b(x)u_x+c(x)u$. $\endgroup$
    – thnguyen
    Jun 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: $\Vert u \Vert_{H^2(\Omega)} \leq C\left( \Vert f\Vert_{L^2(\Omega)}+\Vert u\Vert_{L^2(\Omega)}\right)$ $\endgroup$
    – thnguyen
    Jun 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. $\endgroup$ Jun 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 $f\in L^2(\Omega)$. $\endgroup$
    – thnguyen
    Jun 22, 2018 at 15:55

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.