Correct Sobolev Space for a Boundary Value Problem

Question

Given the two-point BVP: \begin{align} -(a(x)u'(x))'=f,\text{ in }(0,1)\\ u(0)=0\\ u'(1)=0 \end{align}

What is the appropriate Hilbert space for this problem. My idea is $H=H^2(0,1)\cap H_0^1[0,1]$. Is this correct?

Answer 1

Let $V=\{v\in H^1(0,1)\ |\ v(0)=0\}$. By multiplying the equation by some $v\in V$ and integrating on $(0,1)$, the problem becomes: find $u\in V$ such that $$ \int_0^1 a u' v' = \int_0^1 f v \qquad \forall v\in V. $$ Suppose that $a(x)$ satisfies $a(x)\ge \alpha$ for all $x\in (0,1)$ and some $\alpha > 0$. Then the problem is well posed. Note that the Neumann BC is natural.