Construction of a function $u$ such that $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ and $u \not\in W_0^{2,2}(\Omega)$

Question

I'm wondering about an example of a function $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ such that $u \not\in W_0^{2,2}(\Omega)$. Clearly $W_0^{2,2}(\Omega) \subset W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ but that's all I have. I believe that some example could be easily constructed using Fourier series, but I'm having a trouble finishing this idea.

Thanks.

Answer 1

An "abstract" construction can be done as follows: Take $u$ the solution of the problem $-\Delta u= 1$ in $\Omega$ and $u=0$ on $\partial \Omega$. If $\Omega$ is smooth enough then $u\in H^2(\Omega)$ by elliptic regularity. On the other hand, each partial derivative is harmonic so if $u\in H_0^2(\Omega)$, this would mean that the partial derivatives have zero boundary values which, by uniqueness in the Dirichlet problem, implies $\nabla u=0$ which means $u$ is constant, a contradiction.

With this idea you can give an explicit counterexample in the unit ball given by $u(x)=1-|x|^2$. A related counterexample can be given in the cube $[-1,1]^n$ with $v(x)=\prod_{i=1}^n (1-x_i^2)$.