I've been given:
$$−εu′′ + bu′ = f ,x ∈ (0, 1), u(0) = u(1) = 0$$
and have been asked to prove the regularity estimate $$∥u′′∥_{L^2(0,1)} ≤ C_R∥f∥_{L^2(0,1)}$$ I normally try to provide some working for a question when I ask it, I have however made no concrete progress in this problem. I know by what I've done already the weak formulation of this problem and that there exists a solution to this weak formulation, however I don't know how this could help. I started by working on the error term however I feel like I may be going in the wrong direction as rearranging the given equation doesn't involve my bilinear or linear forms. Where is a good place to start, I feel very stuck.
So you have : $$\varepsilon\int_0^1 |u''(x)|dx = \int_0^1 |f(x)-bu'(x)|dx\leq \|f\|_2 + b\|u'\|_2 $$ and you just now need to obtain a bound: $$\|u'\|_2\leq C\|f\|_2.$$
Can you attempt a similar idea and see what happens ? It will probably involve by integrating once and get rid of $u''$ first and use the boundary vanishing condition $u(1) = u(0) = 0.$