Sobolev function in $X = W_0^{1,2}(\Omega) \cap W^{2,2}(\Omega) $

Question

Let $\Omega =\left(0,1 \right)$. Given a function $u \in X = W_0^{1,2}(\Omega) \cap W^{2,2}(\Omega) $ I am looking for a function $v \in X$, such that $$v'' =max(0,u'')$$.

Can one always find such a function ? I am a little confused about this, would appreciate any help/hints.

Answer 1

Define $v$ as weak solution of $v''=\max(0,u'')$ in $H^1_0(\Omega)$. Due to Sobolev regularity $v\in H^2(\Omega)$.