Weak formulation with non homogeneous Dirichlet

Question

I have to find the Weak formulation oh this problem:

$$ \left\{ \begin{gathered} u'' = f{\text{ on }} \Omega =\left] {0,1} \right[ \hfill \\ u(0) = \alpha \hfill \\ u(1) = \beta \hfill \\ \end{gathered} \right.$$

Which space should I take?

Answer 1

As you only have Dirichlet boundary conditions, you can take the space $$V=\{v \in H^1(0,1) \mid v(0)=0=v(1)\}$$

You have to use a "lift" function $R$: define $\tilde u = u - R$ so that $\tilde u(0)=0, \tilde u(1)=0$ i.e. $\tilde u \in V$.

You can take $R(\cdot)$ to be an affine map, for instance $R(x)=(\beta - \alpha)x + \alpha$

Your weak formulation will look like this:

$u=\tilde u + R$, and $a(\tilde u, v) := \int_0^1 -\tilde u'v' = \int_0^1 (f \cdot v + R' \cdot v') =: \tilde F(v)$ for all $v\in V$.