Regularity of Nemitskii maps on Sobolev spaces

Question

Let $\Omega\subset \mathbb R^N$ be a bounded smooth domain, and $f\colon\Omega\times \mathbb R\to \mathbb R$ be a smooth function (let's say $C^2$).

Let $X=W^{1,p}(\Omega)$ with $p>1$ be the Sobolev space of $L^p(\Omega)$ functions with first derivatives in $L^p(\Omega)$.

Which conditions on $f$ and $p$ guarantee that the Nemitskii map $F$ defined by $$[F(u)](x)=f(x,u(x))$$ is of class $C^2$ from $X$ to $L^1(\Omega)$ (or $L^q(\Omega)$)?

Do you have some references?

Answer 1

Welcome to MathOverflow. A good reference for such a question is

Goldberg, H.; Kampowsky, W.; Tröltzsch, F., On Nemytskij operators in $L^p$-spaces of abstract functions, Math. Nachr. 155, 127-140 (1992). ZBL0760.47031.

(In fact, the paper is even about Nemytskii operators between vector valued [Bochner] Lebesgue spaces, but of course one recovers the results for real-valued spaces.)

Since the Nemytskii operator acts on the function $u$ only, you would consider $L^r(\Omega)$ such that $W^{1,p}(\Omega) \hookrightarrow L^r(\Omega)$ (Sobolev embedding, e.g. $\frac1r = \frac1p - \frac1N$ if $p < N$) and then use Theorem 9 in the above paper to find out when $F$ is twice continuously differentiable as an operator $L^r(\Omega) \to L^q(\Omega)$.

The conditions would be that ...

• $2 \leq 2q < r < \infty$ or $r = q = \infty$,
• $f$ is twice continuously differentiable with respect to the $\mathbb{R}$ component, say, $y$,
• the partial derivatives $f_y$ and $f_{yy}$ satisfy Carathéodory measurability conditions as functions $\Omega \times \mathbb{R} \to \mathbb{R}$, and finally
• the Nemytskii operator $G$ induced by $f_{yy}$, $$[G(u)](x) := f_{yy}(x,u(x))$$ is continuous as a mapping $L^r(\Omega) \to L^s(\Omega)$, where $s = \frac{rq}{r-2q}$ if $r < \infty$ and $s = \infty$ if $r=\infty$.

For the last condition, there are necessary and sufficient conditions in Section 2 in the paper, depending on finiteness of $q$ (and $s$). For instance, if $q < \infty$, then it is sufficient that $$|f_{yy}(x,y)| \leq \gamma(x) + \beta |y|^{\frac{r}{s}} \quad ((x,y) \in \Omega \times \mathbb{R})$$ for some $\gamma \in L^s(\Omega)$ and $\beta \geq 0$. (See also Remark 7 in the paper.)