Let $p>1$ and consider the function $$\phi(t):=\int_{\Omega}|f+t\,g|^{p},$$ where $f,g\in L^{p}({\Omega})$, and $\Omega$ is a Lebesgue measurable subset of $\mathbb{R}^{n}$. It is known that $\phi$ is differentiable on $\mathbb{R}$ with $$\phi^{\prime}(t):=\frac{p}{2}\int_{\Omega}|f+t\,g|^{p-2} ((f+tg)\bar{g}+(\bar{f}+t\bar{g})g).$$ (See for example "Analysis" of Lieb and Loss).
Is $\phi$ continuously differentiable ? Is it at least continuously differentiable at $t=0$ ?
It is easy to show that $\phi\in C^{1}$, by Holder's inequality that allows passing the limit inside the integral that defines $\phi^{\prime}$.
My question is: Does $\phi^{\prime\prime}$ exist ? Is it continuous ?
There is no reason for that function to be infinitely differentiable. The scalar function $x\mapsto |x|^p$ already fails that property, unless $p$ is an even integer. For example, $$ \frac{d^2}{dx^2} |x|^3= 6|x|, $$ so $|x|^3$ is differentiable only two times.