Weak formulation 1-D PDE with non-homogenous robin boundary condition

Question

Question: Question statement

Worked Solution: Worked solution

I have a few questions I hope you can help me with. Firstly, is my weak formulation correct? Why can't I use a test function in $H^1_0$ space and use a lift function with the Robin b.c.?

Secondly, how do I prove continuity of B and F?

Thanks for reading.

Answer 1

If $u\in H^1_0$, $u$ has zero value on the all boundary. However, in your problem, you have $u$ with non-zero boundary. That's why you cannot use a test function in $H^1_0$.

I think you can refer this post.