Let $\Delta_0$ be the operator $$ \Delta_0\colon f\in C_C^\infty(0,1)\mapsto -f''\in C_C^\infty(0,1)\,. $$ $\Delta_0$ is a symmetric operator of $L^2([0,1])$ and it's closure is $\Delta_{\min}\colon H^2_0([0,1])\to L^2([0,1])$. The adjoint of $\Delta_{\min}$ is denoted by $\Delta_{\max}$ and it's the distributional laplacian.
I know that $H^2([0,1])\subset \text{Dom}(\Delta_{\max})$. How can I prove that the domain of $\Delta_{\max}$ is $H^2([0,1])$? Could you give me some hints or references?
Thanks in advance.