Sturm-Liouville problem: where did we use $p' \in C(\bar{I})$?

Question

I'm reading section 8.4 Some Examples of Boundary Value Problems in Brezis' Functional Analysis.

Example 2 (Sturm-Liouville problem). Consider the problem $$ (18) \quad \left\{\begin{array}{l} -\left(p u^{\prime}\right)^{\prime}+q u=f \quad \text { on } I=(0,1), \\ u(0)=u(1)=0, \end{array}\right. $$ where $p \in C^1(\bar{I}), q \in C(\bar{I})$, and $f \in L^2(I)$ are given with $$ p(x) \geq \alpha>0 \quad \forall x \in I . $$ If $u$ is a classical solution of (18) we have $$ \int_I p u^{\prime} v^{\prime}+\int_I q u v=\int_I f v \quad \forall v \in H_0^1(I) . $$ We use $H_0^1(I)$ as our function space and $$ a(u, v)=\int_I p u^{\prime} v^{\prime}+\int_I q u v \tag{$*$} $$ as symmetric continuous bilinear form on $H_0^1$. If $q \geq 0$ on $I$ this form is coercive by Poincaré's inequality (Proposition 8.13). Thus, by Lax-Milgram's theorem, there exists a unique $u \in H_0^1$ such that $$ (19) \quad a(u, v)=\int_I f v \quad \forall v \in H_0^1(I). $$ Moreover, $u$ is obtained by $$ \min _{v \in H_0^1(I)}\left\{\frac{1}{2} \int_I\left [ p (v^{\prime})^2+q v^2\right]-\int_I f v\right\} . $$ It is clear from (19) that $p u^{\prime} \in H^1$; thus (by Corollary 8.10) $u^{\prime}=(1 / p)\left(p u^{\prime}\right) \in H^1$ and hence $u \in H^2$. Finally, if $f \in C(\bar{I})$, then $p u^{\prime} \in C^1(\bar{I})$, and so $u^{\prime} \in C^1(\bar{I})$, i.e., $u \in C^2(\bar{I})$. Step D carries over and we conclude that $u$ is a classical solution of (18).

We have $q \in C(\bar{I})$ implies $q \in L^k (I)$ for any $k \in [1, \infty]$. Then the integral $\int_I q u v$ in $(*)$ is well-defined. Similarly, $p \in C(\bar{I})$ implies the integral $\int_I p u' v'$ in $(*)$ is well-defined.

Could you explain where we use $p \in C^1(\bar{I})$ or more precisely $p' \in C(\bar{I})$?

Thank you so much for your elaboration!

Answer 1

We use the hypothesis $p' \in C(\bar{I})$ in

thus (by Corollary 8.10) $u^{\prime}=(1 / p)\left(p u^{\prime}\right) \in H^1$.

Recall that

Corollary 8.10 Let $I$ be a (possibly unbounded) open interval of $\mathbb R$. Let $u, v \in W^{1, p}(I)$ with $1 \leq p \leq$ $\infty$. Then $u v \in W^{1, p}(I)$ and $(u v)^{\prime}=u^{\prime} v+u v^{\prime}$. Furthermore, the formula for integration by parts holds: $$ \int_y^x u^{\prime} v=u(x) v(x)-u(y) v(y)-\int_y^x u v^{\prime} \quad \forall x, y \in \bar{I} . $$

To apply Corollary 8.10, we need $1/p \in H^1(I)$. Hence the condition $p' \in C(\bar{I})$ is needed.