TITLE: Correct Sobolev Space for a Boundary Value Problem
QUESTION [0 upvotes]: 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?

REPLY [1 votes]: 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.
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