What is the Intersection between $H^2$ and $H^1_0$?

Question

I'm currently following an introductory course on Finite Element Methods, where we make extensive use of some Sobolev spaces to ensure regularity on our approximate solutions.

For a given domain, $\Omega$, we have the following defined (which I believe to be pretty standard):

• $L_2(\Omega) := \{u : \int_\Omega u^2 < \infty\}$
• $H^m(\Omega) := \{u \in L_2(\Omega) : D^\alpha u \in L_2(\Omega), \forall |\alpha| \leq m \}$ ($\alpha$ is the multi-index for taking all combinations of derivatives)
• $H^m_0(\Omega) := \{u \in H^m(\Omega) : u|_{\partial \Omega} = 0\}$

Within the course and in some other courses I've found the notes for online, I've noticed that $H^2(\Omega) \cap H^1_0(\Omega)$ has been used... But what's the difference between that and $H^2_0(\Omega)$?

Is there a difference that I'm overlooking, an inconsistency on whoever wrote the notes, or do I have my definitions of the function spaces wrong?

Answer 1

As pointed out by gerw, this answer uses the standard definition of $H^m_0$: $$ H^m_0 = \text{closure of } C_c^\infty(\Omega) \text{ wrt to $H^m$-norm} $$ Functions in $H^2\cap H^1_0$ have zero boundary values, while for functions in $H^2_0$ the function itself and its first derivatives are zero on the boundary.