Poisson PDE solution regularity

Question

I want to know the required conditions for the solution of the Poisson equation to exist in $H^2(\Omega)$. I am considering two cases.

Let $\Omega\subset[-1,1]^d$ be a bounded set with

1) $\partial\Omega$ being $C^2$.

2) $\partial\Omega$ is Lipschitz.

Let $u$ solve the Poisson PDE,

$\triangle u=f$

$\gamma u=0$

$\gamma$ is the trace operator. What conditions on $f$ provide that $u\in H^2(\Omega)$ for 1 and 2. Thanks for the help.

Answer 1

For $f\in L^2(\Omega)$:

In 1), $u\in H^2(\Omega)$.

In 2), $u\in H^2_{\text{loc}}(\Omega)$.

See for instance the following paper and the references therein:

G. Savaré. Regularity results for elliptic equations in Lipschitz domains. J. Funct. Anal. 152 (1998), no. 1, 176--201.