4
$\begingroup$

I am having troubles to justify well some inequalities related with the classical theory of elliptic equations. Let us consider the problem \begin{align*} -\Delta c & =f,\text{ in }\Omega\\ \frac{\partial c}{\partial n} & =0,\text{ on }\partial\Omega. \end{align*} on a bounded smooth domain $\Omega,$ satisfying \begin{align*} \int_\Omega cdx & =\int_{\Omega}f=0. \end{align*} My question is, if the inequality \begin{align*} \left\Vert c\right\Vert _{W^{2,p}(\Omega)}\leq C\left\Vert f\right\Vert _{L^{p}(\Omega)} \end{align*} holds for all $p>1$ and what happens when $p=1,$ e.g. Can I say something about.$\left\Vert \nabla c\right\Vert _{L^{\infty}(\Omega)}$ in the two dimensional case? I would highly appreciate some precise reference.

Extra: In Grisvard's book there is a similar related result (Th. 2.4.1.3) that applies for \begin{align*} -\Delta c+ac & =f,\text{ in }\Omega\\ \frac{\partial c}{\partial n} & =0,\text{ on }\partial\Omega. \end{align*} as far as $"a"$ is larger than some positive constant $\lambda_{\Omega}$ and the conclusion is \begin{align*} \left\Vert c\right\Vert _{W^{2,p}(\Omega)}\leq C\left( \left\Vert c\right\Vert _{L^{p}(\Omega)}+\left\Vert f\right\Vert _{L^{p}(\Omega)}\right) \text{ for }p>1\text{.}% \end{align*} Therefore my question is also "What happens if $a>0$ is an arbitrary positive constant?"

$\endgroup$
0

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.