Consider the problem $$-eu''+xu'+u=f$$ $x$ is defined on the interval $I=[0,L]$. $u(0)=u'(L)=0$ where $e > 0$ is a constant. Prove that the solution satisfies $||eu''||≤ ||f||$ where norm is the $L_2$-norm on $I$.
Hi everyone, I'm new to study of pure mathematics hence very inexperienced in proofs. I tried to use triangle inequality or Cauchy-Schwartz inequality in many ways, I tried to write the integrals explicitly but nothing has helped. How should we proceed? What is the strategy at least? Thanks.
This question is the exercise 2.9 of "M. G. Larson, F. Bengzon, The Finite Element Method: Theory, Implementation, and Applications, Springer, 2014 "