$H^2$ and $C^1$ - Sobolev space

Question

Let $\Omega$ be a domain of $\mathbb{R}^d$, $u\in H^2(\Omega)$ and let $k$ be a constant. Can we tell that $k \nabla u \in C^1$n with $u$ solution of the problem $$-\mathrm{div}(k\nabla u)=f \quad \mbox{in } \Omega, u=0 \quad \mbox{on } \partial \Omega,$$ $f\in L^2(\Omega)$?

Answer 1

Assume that $\Omega$ is bounded. Consider the problem

$$\tag{P} \left\{ \begin{array}{ccc} -\Delta u=f &\mbox{ in $\Omega$} \\ u\in H_0^1(\Omega)\cap H^2(\Omega) &\mbox{ in $\partial\Omega$} \end{array} \right. $$

Because $k$ is constant, to answer your question is equivalently to know if $u$ in the problem (P) satisfies $\nabla u\in C^1(\Omega)^n$. For each $f\in L^2(\Omega)$, we can define $S(f)=u\in H_0^1(\Omega)\cap H^2(\Omega)$, where $u$ is the unique solution of (P).

Now the problem is: For which types of $\Omega$ and $f$, we have that $H_0^1(\Omega)\cap H^2(\Omega)\subset C^1(\Omega)$? or even better $H_0^1(\Omega)\cap H^2(\Omega)\subset C^1(\overline{\Omega})$?.

For example, if $n=1$ and $f\in C(\overline{\Omega})$, it can be proved that $u\in C^2(\overline{\Omega})$ (see Brezis PDE book chapter 8 and for more general results, take a look in chapter 9).

On the other hand, if $f$ is only $L^2(\Omega)$, you can construct examples of functions satisfying (P) with $\nabla u$ not in $C^1$. I will leave to you the task to construct such example.

Remark 1: To construct a example, try to explore functions defined by various sentences, like $|x|$, and partition of unity.