**Lemma A.5.** Let {$f_\alpha$}$_{\alpha>0}$ ⊂ $L^p(Q_T)$, $1 \le p < \infty$ be a sequence that converges to $f$ strongly in $L^p(Q_T)$. Then {$T_\alpha(f_\alpha)$}$_{\alpha>0}$ converges to $f$ in $L^p(Q_T)$.

*Proof.* We first notice that, as $f \in L^p(Q_T)$, $|f| < \infty$ a.e. in $Q_T$. By the inverse Lebesgue theorem, we can extract from {$f_\alpha$}$_{\alpha>0}$ a subsequence, still denoted by {$f_\alpha$}$_{\alpha>0}$, such that

$$f_{\alpha} \xrightarrow{\alpha \to \infty} f \text{ a.e. in } Q_T,$$

and there exists $G \in L^p(Q_T)$ such that $|f_\alpha| \le G$ a.e. in $Q_T$. Let $z \in \{|f| < \infty\}$. Then there exists $\alpha_0 > 0$ such that $\forall \alpha \ge \alpha_0$, $|f_\alpha(z)| \le 1+|f(z)| = \alpha_1$. Therefore, for $\alpha \ge \sup(\alpha_0, \alpha_1)$,

$$T_{\alpha}(f_{\alpha})(z) = f_{\alpha}(z) \xrightarrow{\alpha \to \infty} f(z).$$

Consequently, {$T_\alpha(f_\alpha)$}$_{\alpha>0}$ converges a.e. to $f$ in $Q_T$. As we have

$$|T_{\alpha}(f_{\alpha})| \le |f_{\alpha}| \le G \in L^{p}(Q_{T}),$$

the conclusion follows from the Lebesgue theorem, and the whole sequence converges because of the uniqueness of the limit. $\square$

# B Stampacchia Theorem

This appendix is devoted to providing a complete proof of a lighter version of the Stampacchia Theorem, which is nonetheless sufficient to obtain estimates for equations with right-hand sides in $L^1$ or measures thanks to the Boccardo-Gallouët lemma, and to work with renormalized solutions.

Let $\Omega$ denotes a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary. We first state the theorem before turning to the proof.

## B.1 Statement

**Theorem B.1 (Stampacchia).** Let $G : \mathbb{R} \to \mathbb{R}$ be a K-Lipschitz function such that $G(0) = 0$, and let $1 \le p < \infty$. We assume that $G'$ has only finitely many discontinuities of first kind $^2$. Let $u \in W_0^{1,p}(\Omega)$, then $G \circ u \in W_0^{1,p}(\Omega)$ and we have

$$(B.1) \qquad \nabla(G(u)) = G'(u)\nabla u \quad \text{in } \mathcal{D}'(\Omega).$$

**Remark B.1.** This result is well known when $G$ is of class $C^1$; see [10].

**Remark B.2.** One might object that the right-hand side of the last equality does not make sense where $G'(u)$ is not defined (consider, for instance, the case where $G$ is the absolute value and $u = 0$). The proof will assign a meaning to this equality by showing that $\frac{\partial u}{\partial x_i} = 0$ at points where $u$ belongs to the set of discontinuities of $G'$.

**Remark B.3.** As $G$ is a Lipschitz function, there exist $C, D > 0$ such that $\forall x \in \mathbb{R}$, $|G(x)| \le C|x| + D$. Therefore, since $\Omega$ is bounded, for any $u \in L^p(\Omega)$, we have $G(u) \in L^p(\Omega)$. Moreover, by Rademacher's Theorem, $G' \in L^\infty(\mathbb{R})$, hence proving Formula (B.1) is enough to show that $\forall u \in W_0^{1,p}(\Omega)$, $G(u) \in W_0^{1,p}(\Omega)$. Therefore, since $G(0) = 0$, $\forall u \in W_0^{1,p}(\Omega)$, we have $G(u) \in W_0^{1,p}(\Omega)$. Thus, we just have to prove Formula (B.1).

<footnote>$^2$If $a$ is a point of discontinuity of $G'$, then $G'$ has a right-hand and a left-hand limit at $a$</footnote>
## B.2 Reduction to the case $G(x) = |x|$.

We show in this section how one can reduce the problem to the case $G(x) = |x|$. Let $a_1, \dots, a_n$ be the points of discontinuity of $G'$, and let

$$ \ell_{i,l} = \lim_{x \to a_i} G'(x), \quad \ell_{i,r} = \lim_{x > a_i} G'(x). $$

Notice that, according to the assumptions on $G$, $\ell_{i,l} \neq \ell_{i,r}$. For a given $i = 1, \dots, n$, let $h_i$ denote the piecewise affine function defined by

$$ h_i(x) = \begin{cases} \ell_{i,l}(x - a_i) + G(a_i) & \text{if } x \le a_i, \\ \ell_{i,r}(x - a_i) + G(a_i) & \text{if } a_i < x, \end{cases} $$

and consider the function $F$ defined by

$$ F(x) = G(x) - \sum_{i=1}^{n} h_i(x). $$

It is easily checked that $F$ is of class $C^1$, so that for any $u \in W^{1,p}(\Omega)$, we have $F(u) \in W^{1,p}(\Omega)$ and $\nabla(F(u)) = F'(u)\nabla u$. Therefore, by writing

$$ G(x) = F(x) + \sum_{i=1}^{n} h_i(x), $$

the problem is reduced to proving Formula (B.1) for a piecewise affine function. Next, let $h(x)$ be the piecewise affine function defined by

$$ h(x) = \begin{cases} \ell_1(x - a) + \alpha & \text{if } x \le a, \\ \ell_2(x - a) + \alpha & \text{if } a < x, \end{cases} $$

where $\ell_1 \neq \ell_2$. Let $\tilde{h}(x)$ be given by

$$ \tilde{h}(x) = \begin{cases} \ell_1 x & \text{if } x \le 0, \\ \ell_2 x & \text{if } 0 < x, \end{cases} $$

so that

$$ h(x) = \tilde{h}(x - a) + \alpha. $$

Therefore, it remains to prove Formula (B.1) for the function $\tilde{h}$. But we remark that

$$ \forall x \in \mathbb{R}, \quad \tilde{h}(x) = \frac{\ell_2 - \ell_1}{2} |x| + \frac{\ell_1 + \ell_2}{2} x, $$

so the problem is finally reduced to proving Formula (B.1) when $G(x) = |x|$.

## B.3 Proof of the result when $G(x) = |x|$

We prove in this section Formula (B.1) when $G$ is the absolute value, which satisfies

$$ G'(u) = \operatorname{sgn}(u). $$

We first approximate $G$ by

$$ G_{\varepsilon}(x) = \sqrt{x^2 + \varepsilon^2}, $$
a $C^1$ function. From the standard case, for every $\varepsilon > 0$, $\varphi \in \mathcal{D}(\Omega)$, and $i = 1, \dots, N$, we have

$$ (B.2) \qquad \int_{\Omega} G_{\varepsilon}(u) \frac{\partial \varphi}{\partial x_i} = - \int_{\Omega} G'_{\varepsilon}(u) \frac{\partial u}{\partial x_i} \varphi. $$

We now pass to the limit in (B.2) as $\varepsilon \to 0$. On the one hand, we have

$$ G_{\varepsilon}(u) \to |u| \in L^p(\Omega), $$

and moreover

$$ |G_{\varepsilon}(u)| \le C(|u| + 1) \in L^p(\Omega). $$

Since $\varphi \in \mathcal{D}(\Omega)$, in particular $\frac{\partial \varphi}{\partial x_i} \in L^{p'}(\Omega)$. Therefore, by the Lebesgue theorem,

$$ (B.3) \qquad \int_{\Omega} G_{\varepsilon}(u) \frac{\partial \varphi}{\partial x_i} \underset{\varepsilon \to 0}{\longrightarrow} \int_{\Omega} |u| \frac{\partial \varphi}{\partial x_i}. $$

On the other hand,

$$ G'_{\varepsilon}(u) = \frac{u}{\sqrt{u^2 + \varepsilon^2}} \to \chi_{u \neq 0} \operatorname{sgn}(u) \quad \text{everywhere in } \Omega, $$

and

$$ |G'_{\varepsilon}(u)| \le 1 \in L^{\infty}(\Omega). $$

Since $\frac{\partial u}{\partial x_i} \in L^p(\Omega)$ and $\varphi \in L^{p'}(\Omega)$, again by the Lebesgue theorem,

$$ (B.4) \qquad \int_{\Omega} G'_{\varepsilon}(u) \frac{\partial u}{\partial x_i} \varphi \underset{\varepsilon \to 0}{\longrightarrow} \int_{\Omega} \chi_{u \neq 0} \operatorname{sgn}(u) \frac{\partial u}{\partial x_i} \varphi. $$

Combining (B.2), (B.3) and (B.4), we obtain

$$ (B.5) \qquad \frac{\partial G(u)}{\partial x_i} = \frac{\partial |u|}{\partial x_i} = \chi_{u \neq 0} \operatorname{sgn}(u) \frac{\partial u}{\partial x_i} \quad \text{in } \mathcal{D}'(\Omega), $$

To get formula (B.1), it remains to show that

$$ (B.6) \qquad \frac{\partial u}{\partial x_i} = 0 \quad \text{a.e. on } \{u = 0\}, $$

which will remove the factor $\chi_{u \neq 0}$. To this end, we now approximate the absolute value by

$$ \tilde{G}_{\varepsilon}(x) = \sqrt{(x + \varepsilon)^2 + \varepsilon^2}. $$

By the same arguments as above and straightforward calculations, we obtain in $\mathcal{D}'(\Omega)$, hence in $L^p(\Omega)$ and a.e:

$$ (B.7) \qquad \frac{\partial G(u)}{\partial x_i} = \frac{\partial |u|}{\partial x_i} = \left( \chi_{u \neq 0} \operatorname{sgn}(u) + \frac{1}{\sqrt{2}} \chi_{u=0} \right) \frac{\partial u}{\partial x_i}. $$

Combining (B.5) with (B.7) proves that (B.6) holds as well, which concludes the proof.

**Remark B.4.** Following the steps of the proof, one sees that Stampacchia's theorem is also valid for $W^{1,p}(\Omega)$ instead of $W_0^{1,p}(\Omega)$.
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