I've seen that when we deal with Poisson equation with homogeneous boundary conditions, let's say in 2D with a convex domain $\Omega$, we write that the regularity of $u$ is $H^2 \cap H_0^1(\Omega)$. Why can't we just say that it is in $H^2_0(\Omega)$?
Does it have to do with trace theorem?
Consider the equation $$ \begin{cases} -\Delta u =2 & \text{ in } B_1,\\ \quad \, u=0 & \text{ on }\partial B_1, \end{cases} $$ where $B_1$ denotes the unit ball of $\mathbb{R}^n$ with $n\geq 1$. Then the function $u(x)= 1-|x|^2$ is a solution. You can check directly that $u\in H^1_0(B_1)\cap H^2(B_1)$ (or nuke flies and appeal to the regularity theory), but $u\notin H^2_0(B_1)$ since the gradient $\nabla u(x) = -2x$ is not in $H^1_0(B_1)$.