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. 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?"



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