Let $\Omega\subset \mathbb{R}^2$ be a bounded and smooth domain.
Is it true that if some function $f\in H^1(\Omega)$ admits in the distributional sense $\Delta f\in L^2(\Omega)$ then there exists $\dfrac{\partial^2 f}{\partial x_1^2},\dfrac{\partial^2 f}{\partial x_2^2}\in L^2(\Omega)$?
By $\Delta f=g\in L^2(\Omega)$ in $\Omega$ in the distributional sense we mean that $$\int_{\Omega} f(x)\Delta \phi(x)\ dx=\int_{\Omega} g(x)\phi(x)\ dx,\ \forall\ \phi\in C^{\infty}_{c}(\Omega)$$
My feeling is that this is not true, but I cannot find a counterexample yet.