Homogenization of a nonlinear elliptic system with a transport term in L 2

Abstract

We consider the homogenization of a non-linear elliptic system of two equations related to some models in chemotaxis and flows in porous media. One of the equations contains a convection term where the transport vector is only in $L^{2}$ and a right-hand side which is only in $L^{1}$ . This makes it necessary to deal with entropy or renormalized solutions. The existence of solutions for this system has been proved in reference (Comm. Partial Differential Equations 45(7) (2020) 690–713). Here, we prove its stability by homogenization and that the correctors corresponding to the linear diffusion terms still provide a corrector for the solutions of the non-linear system.

Keywords

Nonlinear elliptic system convection term homogenization entropy solutions corrector

1. Introduction

This paper is devoted to study the homogenization of a nonlinear elliptic system in $R^{N}$ containing a transport term which is only in $L^{2}$ . Namely, for a bounded open set $Ω \subset R^{N}$ , $N ⩾ 2$ , two sequences of matrix functions $A_{ε}$ , $M_{ε}$ and a sequence of Carathéodory functions $E_{ε} : Ω \times R \to R$ , we consider the nonlinear system $\begin{matrix} (1.1) & \{\begin{matrix} - div (A_{ε} \nabla u_{ε}) + a_{ε} u_{ε} = - div (u_{ε} M_{ε} \nabla ψ_{ε}) + f_{ε} & in Ω, \\ - div (M_{ε} \nabla ψ_{ε}) = E_{ε} (x, u_{ε}) + g_{ε} & in Ω, \\ u_{ε} = ψ_{ε} = 0 & on \partial Ω . \end{matrix} \end{matrix}$ Here, $A_{ε}$ and $M_{ε}$ denote two sequences of uniformly bounded and uniformly elliptic matrix functions. The sequence $a_{ε}$ is bounded in $L^{1} (Ω)$ and uniformly positive. With respect to the non-linear term $E_{ε} (x, u_{ε})$ , we assume that it grows at infinity as $| u_{ε} |^{θ}$ , with $θ < 2 / N$ . This system is related to some models for flows in porous media and chemotaxis [11,12].

The existence of solution for (1.1) has been proved in [5], where it is also obtained an existence result just assuming $θ ⩽ 4 / (N - 2)$ if $N ⩾ 3$ , $θ < \infty$ if $N = 2$ (see also [4] for $θ = 1$ ). However this needs to assume $f_{ε}$ and $g_{ε}$ small (see (2.31)). Here we just assume $f_{ε}$ bounded in $L^{1} (Ω)$ and equi-integrable, and $g_{ε}$ bounded in $L^{p} (Ω)$ for some $p > N / 2$ .

The results in [5] provide a sequence $ψ_{ε}$ which is bounded in the Sobolev space $H_{0}^{1} (Ω)$ and also in $L^{\infty} (Ω)$ . Thus, in the convection term in the first equation of (1.1), the vector $M_{ε} \nabla ψ_{ε}$ is only bounded in $L^{2} {(Ω)}^{N}$ . We recall that the classical theory assumes that the vector in the convection term is in $L^{N} {(Ω)}^{N}$ for $N ⩾ 3$ , $L^{r} {(Ω)}^{N}$ , with $r > 2$ if $N = 2$ .

An existence result for a linear elliptic equation with a convection term where the coefficient vector is just in $L^{2}$ has been obtained in [2]. The solution obtained has very little regularity. Namely, it is only known that $log (1 + | u_{ε} |)$ is in $H_{0}^{1} (Ω)$ . However, in the case of system (1.1), $(u_{ε}, ψ_{ε})$ is in $H_{0}^{1} {(Ω)}^{2}$ if $f_{ε} \in L^{\frac{2 N}{N + 2}} (Ω)$ ( $L^{r} (Ω)$ , $r > 1$ if $N = 2$ ) and $g_{ε} \in L^{p} (Ω)$ for some $p > N / 2$ . In the case where $f_{ε}$ is only in $L^{1} (Ω)$ , which is the case we are interested in, the function $u_{ε}$ only exists in the sense of the entropy or renormalized solutions (see e.g. [1,10]), i.e. it is only in $W_{0}^{1, q} (Ω)$ for every $q < N / (N - 1)$ and its truncate at a level $h > 0$ is in $H_{0}^{1} (Ω)$ for every $h > 0$ . Moreover, in the first equation in (1.1), we need to use test functions which depend on the unknown $u_{ε}$ . We refer to (2.10) for the correct definition of solution that we will use in the present paper.

Our main result shows that if the matrix functions $A_{ε}$ and $M_{ε}$ H-converge to some matrix functions A and M (see [15,16,19]), $a_{ε}$ converges weakly in $L^{1} (Ω)$ to a function a, $E_{ε}$ converges to E in the sense given by (2.7), $f_{ε}$ converges weakly to f in $L^{1} (Ω)$ and $g_{ε}$ converges weakly to g in $L^{p} (Ω)$ for some $p > N / 2$ , then, for a subsequence of ε, the solutions $(u_{ε}, ψ_{ε})$ of (1.1) converge to a solution of the analogous problem with $A_{ε}$ , $M_{ε}$ , $a_{ε}$ , $E_{ε}$ , $f_{ε}$ and $g_{ε}$ replaced by A, M, a, E, f and g. We also show that the classical elliptic correctors corresponding to the matrix functions $A_{ε}$ and $M_{ε}$ still provide a corrector result for $u_{ε}$ and $ψ_{ε}$ .

The homogenization of an elliptic sequence of linear elliptic equations where the source terms are just in $L^{1} (Ω)$ has been first studied in [14]. In [3] we have considered the case of an equation with a convection term $div (E_{ε} u_{ε})$ , where $E_{ε}$ is bounded in $L^{2} {(Ω)}^{N}$ and $div E_{ε}$ is compact in $H^{- 1} (Ω)$ . In this case, we only have that $log (1 + | u_{ε} |)$ is bounded in $H_{0}^{1} (Ω)$ . We also refer to [7] where it is considered a convection term of the form $E_{ε} \cdot \nabla u_{ε} + div (E_{ε} u_{ε})$ with $E_{ε}$ just bounded in $L^{2} {(Ω)}^{N}$ . In this case, contrary to the case considered in the present paper, it appears a new zero order term in the limit equation (see [6,8,9,18] for related results).

2. Statement of the main results

For a bounded open set $Ω \subset R^{N}$ , $N ⩾ 2$ , two sequences of matrix functions $A_{ε}, M_{ε} \in L^{\infty} {(Ω)}^{N \times N}$ and a sequence of functions $E_{ε} : Ω \times R \to R$ , we are interested in the homogenization problem (1.1).

For the sequences $A_{ε}$ , $M_{ε}$ we assume the existence of $α, γ, τ, λ > 0$ such that $\begin{matrix} (2.1) & \begin{array}{l} max {α | ξ |^{2}, γ | A_{ε} ξ |^{2}} ⩽ A_{ε} ξ \cdot ξ, \\ max {τ | ξ |^{2}, λ | M_{ε} ξ |^{2}} ⩽ M_{ε} ξ \cdot ξ, \forall ξ \in R^{N}, a.e. in Ω . \end{array} \end{matrix}$

The sequence $a_{ε} \in L^{1} (Ω)$ satisfies that there exist $ρ > 0$ and $a \in L^{1} (Ω)$ , such that $\begin{array}{l} (2.2) & a_{ε} ⩾ ρ a.e. in Ω, \\ (2.3) & a_{ε} ⇀ a in L^{1} (Ω) . \end{array}$

$E_{ε}$ are Carathéodory functions in the sense $\begin{array}{l} (2.4) & E_{ε} (\cdot, s) is measurable in Ω, \forall s \in R, \\ (2.5) & E_{ε} (x, \cdot) is continuous in R, for a.e. x \in Ω . \end{array}$ Moreover, we assume there exist $κ \in L^{p} (Ω)$ with $p > N / 2$ , $κ ⩾ 0$ a.e. in Ω, $K > 0$ , and $0 < θ < 2 / N$ , such that $\begin{matrix} (2.6) & | E_{ε} (x, s) | ⩽ κ (x) + K | s |^{θ}, \forall s \in R, a.e. x \in Ω, \end{matrix}$ and there exists $E : Ω \times R \to R$ such that for a.e. $x \in Ω$ , every $s \in R$ and every sequence $s_{ε}$ converging to s we have $\begin{matrix} (2.7) & E_{ε} (x, s_{ε}) \to E (x, s) . \end{matrix}$

For the right-hand sides $f_{ε}, g_{ε} : Ω \to R$ let us assume $\begin{matrix} (2.8) & f_{ε} ⇀ f in L^{1} (Ω), g_{ε} ⇀ g in L^{p} (Ω), p > \frac{N}{2} . \end{matrix}$

Remark 2.1.
Assumption (2.7) implies that E also satisfies (2.4), (2.5) and (2.6).

The existence of solutions for problem (1.1) has been proved in [5], where the problem considered is somewhat less general but the extension is straightforward. Due to $f_{ε}$ in $L^{1} (Ω)$ and the lack of regularity of the term $u_{ε} M_{ε} \nabla ψ_{ε}$ the first equation must be understood in the sense of the entropy or renormalized solutions [1,2,10], where the test functions depend on the unknown $u_{ε}$ . Namely, defining the usual truncated function $T_{h} : R \to R$ , $h > 0$ by $\begin{matrix} (2.9) & T_{h} (s) = \{\begin{matrix} - h & if s < - h, \\ s & if - h ⩽ s ⩽ h, \\ h & if s > h, \end{matrix} \end{matrix}$ we say that $(u_{ε}, ψ_{ε})$ is a solution of (1.1) if $\begin{matrix} (2.10) & \{\begin{matrix} u_{ε} \in W_{0}^{1, q} (Ω), \forall q \in [1, \frac{N}{N - 1}), T_{h} (u_{ε}) \in H_{0}^{1} (Ω), \forall h > 0, \\ a_{ε} u_{ε} \in L^{1} (Ω), ψ_{ε} \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω), \\ \int_{Ω} A_{ε} \nabla u_{ε} \cdot \nabla v d x + \int_{Ω} a_{ε} u_{ε} v d x = \int_{Ω} u_{ε} M_{ε} \nabla ψ_{ε} \cdot \nabla v d x + \int_{Ω} f_{ε} v d x, \\ \int_{Ω} M_{ε} \nabla ψ_{ε} \cdot \nabla w d x = \int_{Ω} E_{ε} (x, u_{ε}) w d x + \int_{Ω} g_{ε} w d x, \\ \forall w \in H_{0}^{1} (Ω), \forall v \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω) such that \exists R > 0, v^{-}, v^{+} \in R, \\ v = v^{-} a.e. in {u_{ε} ⩽ - R}, v = v^{+} a.e. in {u_{ε} ⩾ R} . \end{matrix} \end{matrix}$
Remark 2.2.
Taking into account that $\begin{array}{l} \int_{Ω} A_{ε} \nabla u_{ε} \cdot \nabla v d x = \int_{Ω} A_{ε} \nabla T_{R} (u_{ε}) \cdot \nabla v d x, \\ \int_{Ω} u_{ε} M_{ε} \nabla ψ_{ε} \cdot \nabla v d x = \int_{Ω} T_{R} (u_{ε}) M_{ε} \nabla ψ_{ε} \cdot \nabla v d x, \end{array}$ we get that the first and third third integral in the third line of (2.10) have a sense.

On the other hand, we remark that for $N ⩾ 4$ , $u_{ε}$ is not in general in $L^{2} (Ω)$ and thus $u_{ε} M \nabla ψ_{ε}$ is not well defined in the distribution sense because it is not in $L^{1} {(Ω)}^{N}$ . When $f_{ε}$ is in $L^{\frac{2 N}{N + 2}} (Ω)$ , ( $L^{r} (Ω)$ with $r > 1$ if $N = 2$ ), then $u_{ε}$ is in $H_{0}^{1} (Ω)$ but $div (u_{ε} M_{ε} \nabla ψ_{ε})$ is not in $H^{- 1} (Ω)$ .

As a particular case of function v we can take $v = T_{k} (u_{ε} - ϕ)$ with $ϕ \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω)$ . This is the type of test functions which are used in the definition of entropy solutions [1,2]. Another example is given by $v = h (u) φ$ with $h \in W^{1, \infty} (R)$ with compact support, $φ \in H_{0}^{1} (Ω)$ . This is the type of test functions which are used in the definition of renormalized solutions [10].
Remark 2.3.
As we said in the introduction, problem (1.1) is related to some models for porous media and chemotaxis. In these applications $u_{ε}$ and $ψ_{ε}$ represent concentrations and therefore they must be non-negative.

Assuming $f_{ε} ⩾ 0$ a.e. in Ω and $\begin{matrix} (2.11) & E_{ε} (x, s) + g_{ε} (x) ⩾ 0, \forall s \in R, a.e. x \in Ω, \end{matrix}$ we can show that effectively, $u_{ε}$ , $ψ_{ε}$ are non-negative functions. In order to prove this result, we take $v = - T_{h} (u_{ε}^{-})$ in the first equation and $w = T_{h} {(u_{ε}^{-})}^{2} / 2$ in the second one, dividing by h, and then passing to the limit when h tends to zero we get (see Step 1 in the proof of Theorem 2.9 for a similar calculus) $\begin{matrix} \int_{{u_{ε} < 0}} a_{ε} | u_{ε} | d x ⩽ 0 . \end{matrix}$ This proves $u_{ε} ⩾ 0$ a.e. in Ω. The maximum principle for linear elliptic equations and (2.11) then show that $ψ_{ε}$ is also non-negative.

We will not need to have $u_{ε}$ , $ψ_{ε}$ non-negative in the following.

In order to state our main result, we need to recall the main results related to the H-convergence theory [15,19]. We also refer to [16] for related results.
Definition 2.4.
Let Ω be a bounded open set of $R^{N}$ and $Q_{ε} \in L^{\infty} {(Ω)}^{N \times N}$ be such that there exist $r, R > 0$ satisfying $\begin{matrix} (2.12) & max {r | ξ |^{2}, R | Q_{ε} ξ |^{2}} ⩽ Q_{ε} ξ \cdot ξ, \forall ξ \in R^{N}, a.e. in Ω . \end{matrix}$ We say that $Q_{ε}$ H-converges to a matrix function $Q \in L^{\infty} {(Ω)}^{N \times N}$ , which satisfies (2.12) with the same constants r, R, if for every $f \in H^{- 1} (Ω)$ the sequence of solutions $u_{ε}$ of $\begin{matrix} (2.13) & \{\begin{matrix} - div (Q_{ε} \nabla u_{ε}) = f & in H^{- 1} (Ω), \\ u_{ε} = 0 & on \partial Ω, \end{matrix} \end{matrix}$ satisfies $\begin{matrix} (2.14) & u_{ε} ⇀ u in H_{0}^{1} (Ω), Q_{ε} \nabla u_{ε} ⇀ Q \nabla u in L^{2} {(Ω)}^{N}, \end{matrix}$ with u the solution of $\begin{matrix} (2.15) & \{\begin{matrix} - div (Q \nabla u) = f & in H^{- 1} (Ω), \\ u = 0 & on \partial Ω . \end{matrix} \end{matrix}$
Remark 2.5.
We recall that (2.12) is equivalent to $Q_{ε}$ uniformly elliptic and bounded. Namely: If $Q_{ε}$ satisfies (2.12), then $\begin{matrix} | Q_{ε} ξ | ⩽ \frac{1}{R} | ξ |, \forall ξ \in R^{N}, a.e. in Ω . \end{matrix}$ Reciprocally, if there exist $r, T > 0$ such that $\begin{matrix} (2.16) & r | ξ |^{2} ⩽ Q_{ε} ξ \cdot ξ, | Q_{ε} ξ | ⩽ T | ξ |, \forall ξ \in R^{N}, a.e. in Ω, \end{matrix}$ then $Q_{ε}$ satisfies (2.12) with $R = r / T^{2}$ .

It is preferable to write (2.12) instead of (2.16) because it is stable by H-convergence.

In the case where $Q_{ε}$ is symmetric, then Q is also symmetric and the constants r, T in (2.16) are stable by H-convergence.
Remark 2.6.
More generally than the convergence result in the definition of H-convergence, it is known that if $Q_{ε}$ H-converges to Q, and $u_{ε} \in H^{1} (Ω)$ is such that $\begin{matrix} (2.17) & u_{ε} ⇀ u in H^{1} (Ω), div (Q_{ε} \nabla u_{ε}) is compact in H^{- 1} (Ω), \end{matrix}$ then $\begin{matrix} (2.18) & Q_{ε} \nabla u_{ε} ⇀ Q \nabla u in L^{2} {(Ω)}^{N} . \end{matrix}$

The main interest of the above definition are the following compactness and corrector results
Theorem 2.7.
Let Ω be a bounded open set of $R^{N}$ and $Q_{ε}$ be satisfying ( 2.12 ) for some $r, R > 0$ . Then, for a subsequence of ε, still denoted by ε, there exists $Q \in L^{\infty} {(Ω)}^{N \times N}$ such that $Q_{ε}$ H-converges to Q.
Theorem 2.8.
Assume that $Q_{ε}$ H-converges to Q in a bounded open set $Ω \subset R^{N}$ . Taking ${e_{1}, \dots, e_{N}}$ the chanonical basis of $R^{N}$ , and $p_{ε}^{i}$ , $1 ⩽ i ⩽ N$ , the solution of $\begin{matrix} (2.19) & \{\begin{matrix} - div (Q_{ε} \nabla p_{ε}^{i}) = - div (Q e_{i}) & in Ω, \\ p_{ε}^{i} = x_{i} & on \partial Ω, \end{matrix} \end{matrix}$ we have $\begin{matrix} (2.20) & p_{ε}^{i} ⇀ x_{i} in H^{1} (Ω), p_{ε}^{i} \overset{}{⇀} x_{i} in L^{\infty} (Ω), \end{matrix}$ and for every* $u_{ε} \in H^{1} (Ω)$ which satisfies ( 2.17 ), we have $\begin{matrix} (2.21) & \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} u \nabla p_{ε}^{i} \to 0 in L^{1} {(Ω)}^{N} . \end{matrix}$ Here, $L^{1} {(Ω)}^{N}$ can be replaced by $L^{2} {(Ω)}^{N}$ if u belongs to $H_{0}^{1} (Ω) \cap W^{1, \infty} (Ω)$ .

Our main result in the present paper is given by the following theorem Theorem 2.9.
Let Ω be a bounded open set of $R^{N}$ . We consider two sequences of matrix functions $A_{ε}$ , $M_{ε}$ which satisfy ( 2.1 ) and H-converge to $A, M \in L^{\infty} {(Ω)}^{N \times N}$ respectively, $a_{ε} \in L^{1} (Ω)$ which satisfies ( 2.2 ) and ( 2.3 ), and $E_{ε}$ which satisfies ( 2.4 ), ( 2.5 ), ( 2.6 ) and ( 2.7 ). Then, for every $f_{ε} \in L^{1} (Ω)$ , $g_{ε} \in L^{p} (Ω)$ , $p > N / 2$ , which satisfy ( 2.8 ), and every solution $(u_{ε}, ψ_{ε})$ of ( 1.1 ), we have: $\begin{array}{l} (2.22) & ψ_{ε} is bounded in H_{0}^{1} (Ω) \cap L^{\infty} (Ω), \\ (2.23) & u_{ε} is bounded in W_{0}^{1, q} (Ω), \forall q \in [1, \frac{N}{N - 1}), \\ (2.24) & T_{h} (u_{ε}) is bounded in H_{0}^{1} (Ω), \forall h > 0 . \end{array}$ Taking a subsequence of ε, still denoted by ε, such that there exist ψ, u satisfying $\begin{array}{l} (2.25) & ψ_{ε} ⇀ ψ in H_{0}^{1} (Ω), ψ_{ε} \overset{}{⇀} ψ in L^{\infty} (Ω), \\ (2.26) & u_{ε} ⇀ u in W_{0}^{1, q} (Ω), \forall q \in [1, \frac{N}{N - 1}), T_{h} (u_{ε}) ⇀ T_{h} (u) in H_{0}^{1} (Ω), \forall h > 0, \end{array}$ we have that* $(u, ψ)$ is a solution of $\begin{matrix} (2.27) & \{\begin{matrix} - div (A \nabla u) + a u = - div (u M \nabla ψ) + f & in Ω \\ - div (M \nabla ψ) = E (x, u) + g & in Ω \\ u = ψ = 0 & on \partial Ω, \end{matrix} \end{matrix}$ Moreover, defining $w_{ε}^{i}$ and $z_{ε}^{i}$ , $1 ⩽ i ⩽ N$ , by $\begin{matrix} (2.28) & \{\begin{matrix} - div (A_{ε} \nabla w_{ε}^{i}) = - div (A e_{i}) & in Ω, \\ w_{ε}^{i} = x_{i} & on \partial Ω, \end{matrix} \{\begin{matrix} - div (M_{ε} \nabla z_{ε}^{i}) = - div (M e_{i}) & in Ω, \\ z_{ε}^{i} = x_{i} & on \partial Ω . \end{matrix} \end{matrix}$ we have $\begin{array}{l} (2.29) & lim_{ε \to 0} \int_{{| u_{ε} | ⩽ h}} | \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} T_{h} (u) \nabla w_{ε}^{i} | d x = 0, \forall h > 0 . \\ (2.30) & \nabla ψ_{ε} - \sum_{i = 1}^{N} \partial_{i} ψ \nabla z_{ε}^{i} \to 0 in L^{1} {(Ω)}^{N} . \end{array}$
Remark 2.10.
If $f_{ε}$ in Theorem 2.9 is bounded in $L^{s} (Ω)$ with $\begin{matrix} 1 < s ⩽ \frac{2 N}{N + 2} if N > 2, s > 1 if N = 2, \end{matrix}$ then $u_{ε}$ is bounded in $\begin{matrix} W^{1, \frac{N s}{N - s}} (Ω), if N > 2, H_{0}^{1} (Ω) if N = 2 . \end{matrix}$

We also observe that if $M_{ε} \in C^{0} {(\overline{Ω})}^{N \times N}$ and Ω is a Lipschitz domain, then $ψ_{ε}$ is in $W^{1, p} (Ω)$ . However in homogenization theory, we are interested in strongly oscillating coefficients and therefore assuming $M_{ε}$ compact in $C^{0} {(\overline{Ω})}^{N \times N}$ is not a good assumption.
Remark 2.11.
Following the results in [5], it is still possible to prove the existence of solutions for (1.1) taking $ρ = 0$ in (2.2) and $\begin{matrix} θ < \infty if N = 2, θ ⩽ \frac{4}{N - 2} if N ⩾ 3, \end{matrix}$ but then, for $\begin{matrix} p > 1 if N = 2, p = \frac{N}{2} if N ⩾ 3, q > 1 if N = 2, q = \frac{2 N}{N + 2} if N ⩾ 3 \end{matrix}$ we need to assume $\begin{matrix} (2.31) & {‖ E_{ε} (\cdot, 0) + g_{ε} ‖}_{L^{p} (Ω)} + ‖ f_{ε} ‖_{L^{q} (Ω)} ⩽ δ, \end{matrix}$ for some $δ > 0$ depending on Ω and the constant α in (2.1). With these assumptions and using the same reasoning below, we can show that Theorem 2.7 still holds true, where now $u_{ε}$ is bounded in $H_{0}^{1} (Ω)$ and $ψ_{ε}$ is bounded in $H_{0}^{1} (Ω) \cap L^{r} (Ω)$ for every $r < \infty$ .

3. Proof of the homogenization result

Our aim in this section is to prove Theorem 2.9 which describes the asymptotic behavior of the solutions of problem (1.1). The proof is based on the estimates for (1.1) obtained in [5].

Proof of Theorem 2.9.

First of all we observe that replacing θ by $max {θ, \frac{1}{p}}$ , we can always assume $\begin{matrix} (3.1) & p ⩾ \frac{1}{θ} . \end{matrix}$

Along the proof we will denote by C a non-negative constant which only depends on α, γ, p, θ, N, K and $| Ω |$ . It may change from line to line.

Step 1. Assume two sequences $f_{ε} \in L^{1} (Ω)$ , $g_{ε} \in L^{p} (Ω)$ , which satisfy (2.8) for some $f \in L^{1} (Ω)$ and $g \in L^{p} (Ω)$ . Let us obtain some estimates for the solutions of (1.1).

For $h > 0$ , we take $v = T_{h} (u_{ε})$ in the first equation in (2.10) and $w = T_{h} {(u_{ε})}^{2} / 2$ in the second one. Using (2.1) and (2.6), we get $\begin{array}{l} α \int_{Ω} {| \nabla T_{k} (u_{ε}) |}^{2} d x + \int_{Ω} a_{ε} u_{ε} T_{h} (u_{ε}) d x \\ ⩽ \frac{1}{2} \int_{Ω} M_{ε} \nabla ψ_{ε} \cdot \nabla T_{h} {(u_{ε})}^{2} d x + \int_{Ω} f_{ε} T_{h} (u_{ε}) d x \\ = \frac{1}{2} \int_{Ω} E_{ε} (x, u_{ε}) T_{h} {(u_{ε})}^{2} d x + \frac{1}{2} \int_{Ω} g_{ε} T_{h} {(u_{ε})}^{2} d x + \int_{Ω} f_{ε} T_{h} (u_{ε}) d x \\ (3.2) & ⩽ \frac{1}{2} \int_{Ω} (κ + g_{ε} + K | u_{ε} |^{θ}) T_{h} {(u_{ε})}^{2} d x + \int_{Ω} f_{ε} T_{h} (u_{ε}) d x . \end{array}$ Dividing this inequality by h and taking the limit when h tends to zero, we deduce $\begin{matrix} (3.3) & \int_{Ω} a_{ε} | u_{ε} | d x ⩽ \int_{Ω} | f_{ε} | d x, \end{matrix}$ which combined with (2.2) shows $\begin{matrix} (3.4) & u_{ε}, a_{ε} u_{ε} bounded in L^{1} (Ω) . \end{matrix}$ As a consequence, $| u_{ε} |^{θ}$ is bounded in $L^{\frac{1}{θ}} (Ω)$ . Using also that κ belongs to $L^{p} (Ω)$ , $g_{ε}$ is bounded in $L^{p} (Ω)$ , and $1 / θ, p > N / 2$ , we can apply the ellipticity of the operator combined with the classical Stampacchia’s estimates [17] to get $\begin{matrix} (3.5) & ψ_{ε} bounded in H_{0}^{1} (Ω) \cap L^{\infty} (Ω) . \end{matrix}$

By De Giorgi-Moser theory and Meyers’ theorem [13,17], we also have that for every $ω ⋐ Ω$ open, there exist $r > 1$ and $β \in (0, 1]$ such that $\begin{matrix} (3.6) & ψ_{ε} bounded in W^{1, r} (ω) \cap C^{0, β} (ω), \end{matrix}$ where $ω = Ω$ if Ω is smooth enough.

Let us now return to (3.2). By Holder’s and Young’s inequalities, for every $δ > 0$ , we can estimate the first term on the right-hand side by (we assume $N ⩾ 3$ to simplify, the case $N = 2$ is similar). $\begin{array}{l} \int_{Ω} (κ + g_{ε} + K | u_{ε} |^{θ}) T_{h} {(u_{ε})}^{2} d x \\ ⩽ \int_{Ω} (κ + | g_{ε} | + K | u_{ε} |^{θ}) | u_{ε} |^{\frac{2}{N} - θ} {| T_{h} (u_{ε}) |}^{2 - \frac{2}{N} + θ} d x \\ ⩽ C \int_{Ω} ({(κ + | g_{ε} |)}^{\frac{2}{N θ}} + | u_{ε} |^{\frac{2}{N}}) {| T_{h} (u_{ε}) |}^{2 - \frac{2}{N} + θ} d x \\ ⩽ C {(\int_{Ω} ({(κ + | g_{ε} |)}^{\frac{1}{θ}} + | u_{ε} |) d x)}^{\frac{2}{N}} {(\int_{Ω} {| T_{h} (u_{ε}) |}^{\frac{2 N}{N - 2} (1 - \frac{1}{N} + \frac{θ}{2})} d x)}^{\frac{N - 2}{N}} \\ ⩽ C {(\int_{Ω} ({(κ + | g_{ε} |)}^{\frac{1}{θ}} + | u_{ε} |) d x)}^{\frac{2}{N}} {(\int_{Ω} {| T_{h} (u_{ε}) |}^{\frac{2 N}{N - 2}} d x)}^{\frac{N - 2}{N} (1 - \frac{1}{N} + \frac{θ}{2})} \\ ⩽ \frac{C}{δ^{\frac{2 N}{2 - N θ}}} {(\int_{Ω} ({(κ + | g_{ε} |)}^{\frac{1}{θ}} + | u_{ε} |) d x)}^{\frac{4}{2 - N θ}} + δ^{\frac{2 N}{2 N - 2 + N θ}} {(\int_{Ω} {| T_{h} (u_{ε}) |}^{\frac{2 N}{N - 2}} d x)}^{\frac{N - 2}{N}} . \end{array}$ Taking then δ small enough and using Sobolev’s inequality we deduce from (3.2) $\begin{matrix} (3.7) & \int_{Ω} {| \nabla T_{h} (u_{ε}) |}^{2} d x ⩽ C {(\int_{Ω} ({(κ + | g_{ε} |)}^{\frac{1}{θ}} + | u_{ε} |) d x)}^{\frac{4}{2 - N θ}} + | \int_{Ω} f_{ε} T_{h} (u_{ε}) d x |, \forall h > 0 . \end{matrix}$

Step 2. Since $u_{ε}$ is bounded in $L^{1} (Ω)$ , the classical theory for entropy or renormalized solutions [1,10] allows us to deduce from (3.7) $\begin{array}{l} (3.8) & u_{ε} bounded in W_{0}^{1, q} (Ω), \forall q \in [1, \frac{N}{N - 1}), \\ (3.9) & T_{h} (u_{ε}) bounded in H_{0}^{1} (Ω), \forall h > 0 . \end{array}$ On the other hand, the weak convergence of $f_{ε}$ in $L^{1} (Ω)$ combined with Rellich-Kondrachov’s and Egorov’s theorems imply $\begin{matrix} lim_{ε \to 0} \int_{Ω} f_{ε} T_{h} (u_{ε}) d x = \int_{Ω} f T_{h} (u) d x, \end{matrix}$ and therefore, (3.7) also provides $\begin{matrix} (3.10) & lim_{h \to \infty} \underset{ε \to 0}{lim sup} \frac{1}{h} \int_{Ω} {| \nabla T_{h} (u_{ε}) |}^{2} d x = 0 . \end{matrix}$

Estimates (3.8) and (3.9) combined with (3.5) and (3.6) allow us to deduce the existence of $ψ \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω) \cap C^{0} (Ω)$ and $u \in W_{0}^{1, q} (Ω)$ for every $q < N / (N - 1)$ , with $T_{h} (u) \in H_{0}^{1} (Ω)$ for every $h > 0$ , such that for a subsequence of ε, still denoted by ε, we have $\begin{matrix} (3.11) & \{\begin{matrix} ψ_{ε} ⇀ ψ in H_{0}^{1} (Ω), ψ_{ε} \overset{*}{⇀} ψ in L^{\infty} (Ω), \\ | \nabla ψ_{ε} |^{2} equi-integrable in compact subsets of | Ω |, ψ_{ε} \to ψ in C^{0} (Ω), \\ u_{ε} ⇀ u in W_{0}^{1, q} (Ω), \forall q \in [1, \frac{N}{N - 1}), T_{h} (u_{ε}) ⇀ T_{h} (u) in H_{0}^{1} (Ω), \forall h > 0 . \end{matrix} \end{matrix}$ Using (2.6), (2.7) and Rellich-Kondrachov’s compactness theorem, we deduce in particular that $E_{ε} (x, u_{ε}) + g_{ε}$ strongly converges to $E (x, u) + g$ in $H^{- 1} (Ω)$ . By Remark 2.6, this implies $\begin{matrix} (3.12) & M_{ε} \nabla ψ_{ε} ⇀ M \nabla ψ in L^{2} {(Ω)}^{N}, \end{matrix}$ and then, ψ satisfies $\begin{matrix} (3.13) & - div (M \nabla ψ) = E (x, u) + g in Ω . \end{matrix}$ Statement (2.30) now follows from (2.21).

Step 3. In order to get the equation satisfied by u, let us first prove some strong convergence for $div (A_{ε} \nabla u_{ε})$ .

For $h > 0$ , we define $S_{h} : R \to R$ by $\begin{matrix} (3.14) & S_{h} (s) = \{\begin{matrix} 1 - \frac{| s |}{h} & if | s | ⩽ h, \\ 0 & if | s | > h . \end{matrix} \end{matrix}$ Then, for a sequence $ϕ_{ε} \in H^{1} (Ω) \cap L^{\infty} (Ω)$ such that there exists $ϕ \in H^{1} (Ω)$ with $\begin{matrix} (3.15) & supp (ϕ_{ε}) \subset K \subset Ω compact, ϕ_{ε} ⇀ ϕ in H_{0}^{1} (Ω), ϕ_{ε} \overset{*}{⇀} ϕ in L^{\infty} (Ω), \end{matrix}$ we take $v = S_{h} (u_{ε}) ϕ_{ε}$ and $w = T_{h} (u) S_{h} (u_{ε}) ϕ_{ε}$ in (2.10). This provides $\begin{array}{l} \int_{Ω} A_{ε} \nabla u_{ε} \cdot \nabla ϕ_{ε} S_{h} (u_{ε}) d x + \int_{Ω} A_{ε} \nabla u_{ε} \cdot \nabla u_{ε} S_{h}^{'} (u_{ε}) ϕ_{ε} d x + \int_{Ω} a_{ε} u_{ε} S_{h} (u_{ε}) ϕ_{ε} d x \\ = \int_{Ω} (T_{h} (u_{ε}) - T_{h} (u)) M_{ε} \nabla ψ_{ε} \cdot \nabla (S_{h} (u_{ε}) ϕ_{ε}) d x + \int_{Ω} E_{ε} (x, u_{ε}) T_{h} (u) S_{h} (u_{ε}) ϕ_{ε} d x \\ (3.16) & + \int_{Ω} g_{ε} T_{h} (u) S_{h} (u_{ε}) ϕ_{ε} d x - \int_{Ω} M_{ε} \nabla ψ_{ε} \cdot \nabla T_{h} (u) S_{h} (u_{ε}) ϕ_{ε} d x + \int_{Ω} f_{ε} S_{h} (u_{ε}) ϕ_{ε} d x . \end{array}$ In the second term in this equality, thanks to (3.10), we have $\begin{matrix} (3.17) & lim_{h \to \infty} \underset{ε \to 0}{lim sup} | \int_{Ω} A_{ε} \nabla u_{ε} \cdot \nabla u_{ε} S_{h}^{'} (u_{ε}) ϕ_{ε} d x | = 0 . \end{matrix}$ For the fourth term, we use (3.11) and $S_{h} (u_{ε}) ϕ_{ε}$ bounded in $H_{0}^{1} (Ω)$ , with compact support, which provides $\begin{matrix} (3.18) & lim_{ε \to 0} \int_{Ω} (T_{h} (u_{ε}) - T_{h} (u)) \nabla ψ_{ε} \cdot \nabla (S_{h} (u_{ε}) ϕ_{ε}) d x = 0 . \end{matrix}$ In the remainder terms except the first one, we easily pass to the limit when $ε \to 0$ by (2.3), (2.7), (3.11), Rellich-Kondrachov’s compactness theorem and (3.12). We deduce $\begin{array}{l} lim_{h \to \infty} | lim_{ε \to 0} \int_{Ω} A_{ε} \nabla u_{ε} \cdot \nabla ϕ_{ε} S_{h} (u_{ε}) d x + \int_{Ω} a u S_{h} (u) ϕ d x - \int_{Ω} E (x, u) u S_{h} (u) ϕ d x \\ - \int_{Ω} g u S_{h} (u) ϕ d x + \int_{Ω} M \nabla ψ \nabla u S_{h} (u) ϕ d x - \int_{Ω} f S_{h} (u) ϕ d x | = 0, \end{array}$ which taking into account (3.13) can also be written as $\begin{array}{l} lim_{h \to \infty} | lim_{ε \to 0} \int_{Ω} A_{ε} \nabla u_{ε} \cdot \nabla ϕ_{ε} S_{h} (u_{ε}) d x + \int_{Ω} a u S_{h} (u) ϕ d x \\ (3.19) & - \int_{Ω} u M \nabla ψ \cdot \nabla (S_{h} (u) ϕ) d x - \int_{Ω} f S_{h} (u) ϕ d x | = 0 . \end{array}$

Step 4. Thanks to (3.11), and extracting a subsequence if necessary, we can assume that there exists $σ \in L^{q} {(Ω)}^{N}$ for every $q \in [1, N / (N - 1))$ such that $\begin{matrix} (3.20) & A_{ε} \nabla u_{ε} ⇀ σ in L^{q} {(Ω)}^{N}, 1 ⩽ q < \frac{N}{N - 1} . \end{matrix}$ Moreover, Rellich-Kondrachov’s compactness theorem and (3.11) imply that $σ χ_{{| u | ⩽ h}} \in L^{2} {(Ω)}^{N}$ , for every $h > 0$ and $\begin{matrix} (3.21) & A_{ε} \nabla T_{h} (u_{ε}) ⇀ σ χ_{{| u | ⩽ h}} in L^{2} {(Ω)}^{N}, \forall h > 0 outside a countable set . \end{matrix}$

Let us obtain a variational equation for σ. For this purpose, we take in (3.19) $ϕ_{ε} = ϕ$ with $\begin{matrix} (3.22) & ϕ \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω), supp (ϕ) compact, \exists R > 0 with ϕ = 0 a.e. in {| u | ⩾ R} . \end{matrix}$ Thanks to (3.21), $S_{h} (u)$ converging a.e. to one and bounded in $L^{\infty} (Ω)$ , and $ϕ = 0$ a.e. in ${| u | > R}$ , this proves $\begin{matrix} (3.23) & \int_{Ω} σ \cdot \nabla ϕ d x + \int_{Ω} a u ϕ d x = \int_{Ω} u M \nabla ψ \cdot \nabla ϕ d x + \int_{Ω} f ϕ d x, \end{matrix}$ for every ϕ which satisfies (3.22). By density, the assumption $supp (ϕ)$ compact can be replaced by $ϕ \in H_{0}^{1} (Ω)$ .

Assume now $\begin{matrix} (3.24) & v \in H_{0}^{1} (Ω) \cap L^{\infty} (Ω), \exists R > 0, v^{-}, v^{+} \in R such that \{\begin{matrix} v = v^{-} & a.e in {u ⩽ - R}, \\ v = v^{+} & a.e in {u ⩾ R} . \end{matrix} \end{matrix}$ Then, for every $h > 0$ , we can choose in (3.23) $\begin{matrix} ϕ = v - \frac{1}{h} (v^{-} T_{h} {(u)}^{-} + v^{+} T_{h} {(u)}^{+}) . \end{matrix}$ This gives $\begin{array}{l} \int_{Ω} σ \cdot \nabla v d x + \int_{Ω} a u v d x \\ = \int_{Ω} u M \nabla ψ \cdot \nabla v d x + \int_{Ω} f v d x + \frac{1}{h} \int_{Ω} σ \cdot (v^{-} \nabla T_{h} {(u)}^{-} + v^{+} \nabla T_{h} {(u)}^{+}) d x \\ + \frac{1}{h} \int_{Ω} a u (v^{-} T_{h} {(u)}^{-} + v^{+} T_{h} {(u)}^{+}) d x - \frac{1}{h} \int_{Ω} u M \nabla ψ \cdot (v^{-} \nabla T_{h} {(u)}^{-} + v^{+} \nabla T_{h} {(u)}^{+}) d x \\ (3.25) & - \frac{1}{h} \int_{Ω} f (v^{-} T_{h} {(u)}^{-} + v^{+} T_{h} {(u)}^{+}) d x . \end{array}$ Let us pass to the limit when h tends to infinity in this equality.

For the fifth term we remark that (3.17), (3.21) and $A_{ε}$ bounded in $L^{\infty} {(Ω)}^{N \times N}$ imply $\begin{matrix} (3.26) & lim_{h \to \infty} \frac{1}{h} \int_{{0 < | u | < h}} | \nabla u |^{2} d x = lim_{h \to \infty} \frac{1}{h} \int_{{0 < | u | < h}} | σ |^{2} d x = 0, \end{matrix}$ and thus $\begin{matrix} lim_{h \to \infty} \frac{1}{h} \int_{Ω} σ \cdot (v^{-} \nabla T_{h} {(u)}^{-} + v^{+} \nabla T_{h} {(u)}^{+}) d x = 0 . \end{matrix}$ For the sixth term, an immediate application of Lebesgue dominated convergence theorem gives $\begin{matrix} lim_{h \to \infty} \frac{1}{h} \int_{Ω} a u (v^{-} T_{h} {(u)}^{-} + v^{+} T_{h} {(u)}^{+}) d x = 0 . \end{matrix}$ For the seventh term we use (3.13), which combined with the Sobolev imbedding theorem provides (we assume $N ⩾ 3$ , the case $N = 2$ is similar) $\begin{array}{rcl} \int_{Ω} u M \nabla ψ \cdot \nabla T_{h} {(u)}^{-} d x & = & - \frac{1}{2} \int_{Ω} E (x, u) {| T_{h} {(u)}^{-} |}^{2} d x - \frac{1}{2} \int_{Ω} g {| T_{h} {(u)}^{-} |}^{2} d x \\ ⩽ & \frac{1}{2} {(\int_{Ω} {(| E (x, u) | + | g |)}^{\frac{N}{2}} d x)}^{\frac{2}{N}} {(\int_{Ω} {| T_{h} (u) |}^{\frac{2 N}{N - 2}} d x)}^{\frac{N - 2}{N}} \\ ⩽ & C {(\int_{Ω} {(| E (x, u) | + | g |)}^{\frac{N}{2}} d x)}^{\frac{2}{N}} \int_{Ω} {| \nabla T_{h} (u) |}^{2} d x . \end{array}$ Here we observe that (2.6), $u \in W^{1, q} (Ω)$ for every $q < N / (N - 1)$ , and $g \in L^{p} (Ω)$ , $p > N / 2$ imply that $E (x, u)$ and g belong to $L^{\frac{N}{2}} (Ω)$ . Thus, using (3.26) we get $\begin{matrix} lim_{h \to \infty} \frac{1}{h} \int_{Ω} u M \nabla ψ \cdot \nabla T_{h} {(u)}^{-} d x = 0, \end{matrix}$ and analogously $\begin{matrix} lim_{h \to \infty} \frac{1}{h} \int_{Ω} u M \nabla ψ \cdot \nabla T_{h} {(u)}^{+} d x = 0 . \end{matrix}$ Finally, the last term in (3.25) tends to zero as a simple application of Lebesgue dominated convergence theorem.

Therefore, passing to the limit when h tends to zero in (3.25), we conclude $\begin{matrix} (3.27) & \int_{Ω} σ \cdot \nabla v d x + \int_{Ω} a u v d x = \int_{Ω} u M \nabla ψ \cdot \nabla v d x + \int_{Ω} f v d x, \end{matrix}$ for every v which satisfies (3.24).

Step 5. Let us now prove that $σ = A \nabla u$ which combined with (3.27) provides the first equation in (2.27).

For $m > 0$ , $ϕ \in C^{\infty} (\overline{Ω})$ and $φ \in C_{c}^{\infty} (Ω)$ , we define $v_{ε}$ by $\begin{matrix} (3.28) & v_{ε} = T_{m} (u_{ε} - ϕ - \sum_{i = 1}^{N} \partial_{i} ϕ (w_{ε}^{i} - x_{i})) φ, \end{matrix}$ where the functions $w_{ε}^{i}$ are defined by (2.28) and by Theorem 2.7, satisfy $\begin{matrix} (3.29) & \begin{array}{l} w_{ε}^{i} - x_{i} ⇀ 0 in H_{0}^{1} (Ω), w_{ε}^{i} \overset{*}{⇀} x_{i} in L^{\infty} (Ω), w_{ε}^{i} \to x_{i} in C^{0} (Ω) \\ {| \nabla w_{ε}^{i} |}^{2} equi-integrable in compact subsets of | Ω |, A_{ε} \nabla w_{ε}^{i} ⇀ A e_{i} in L^{2} {(Ω)}^{N} . \end{array} \end{matrix}$

Taking $\partial_{i} ϕ v_{ε} S_{h} (u_{ε}) φ$ as test function in the first equation in (2.28) and adding in i we easily get $\begin{array}{l} lim_{ε \to 0} \sum_{i = 1}^{N} \int_{U_{ε}^{m}} A_{ε} \partial_{i} ϕ \nabla w_{ε}^{i} \cdot (\nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i}) S_{h} (u_{ε}) φ d x \\ (3.30) & = \int_{Ω} A \nabla ϕ \cdot \nabla T_{m} (u - ϕ) S_{h} (u) φ d x, \end{array}$ with $\begin{matrix} U_{ε}^{m} = {x \in Ω : | u_{ε} - ϕ - \sum_{i = 1}^{N} \partial_{i} ϕ (w_{ε}^{i} - e_{i}) | ⩽ m} . \end{matrix}$

On the other hand, using (3.19) with $ϕ_{ε} = v_{ε}$ , (3.21), and that $w_{ε}^{i} - x_{i}$ converges uniformly to zero on compact subsets of Ω, we get $\begin{array}{l} lim_{h \to \infty} | lim_{ε \to 0} \int_{U_{ε}^{m}} A_{ε} \nabla u_{ε} \cdot (\nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i}) S_{h} (u_{ε}) φ d x + \int_{Ω} σ \cdot \nabla φ T_{m} (u - ϕ) S_{h} (u) d x \\ + \int_{Ω} a u S_{h} (u) T_{m} (u - ϕ) φ d x - \int_{Ω} u M \nabla ψ \cdot \nabla (S_{h} (u) T_{m} (u - ϕ) φ) d x \\ - \int_{Ω} f S_{h} (u) T_{m} (u - ϕ) φ d x | = 0, \end{array}$ for every $m > 0$ outside a countable set. Thanks to (3.27), this can be written as $\begin{array}{l} lim_{h \to \infty} | lim_{ε \to 0} \int_{U_{ε}^{m}} A_{ε} \nabla u_{ε} \cdot (\nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i}) S_{h} (u_{ε}) φ d x \\ - \int_{Ω} σ \cdot \nabla T_{m} (u - ϕ) S_{h} (u) φ d x - \int_{Ω} σ \cdot \nabla S_{h} (u) T_{m} (u - ϕ) φ d x | = 0, \end{array}$ or taking into account that $\nabla T_{m} (u - ϕ) = 0$ if $| u | > ‖ ϕ ‖_{L^{\infty} (Ω)} + m$ , $\begin{matrix} (3.31) & lim_{h \to \infty} lim_{ε \to 0} \int_{U_{ε}^{m}} A_{ε} \nabla u_{ε} \cdot (\nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i}) S_{h} (u_{ε}) φ d x = \int_{Ω} σ \cdot \nabla T_{m} (u - ϕ) φ d x . \end{matrix}$ Taking the difference of (3.31) and (3.30) and using that $‖ S_{h} (u_{ε}) - 1 ‖_{L^{\infty} (U_{ε}^{m})}$ tends to zero when h tends to infinity uniformly in ε, we have then proved that $\begin{array}{l} lim_{ε \to 0} \int_{U_{ε}^{m}} A_{ε} (\nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i}) \cdot (\nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i}) φ d x \\ (3.32) & = \int_{Ω} (σ - A \nabla ϕ) \cdot \nabla T_{m} (u - ϕ) φ d x, \end{array}$ for every $m \in N$ , $ϕ \in C^{\infty} (\overline{Ω})$ and $φ \in C_{c}^{\infty} (Ω)$ . Assuming $φ ⩾ 0$ , and taking into account (2.1), this implies $\begin{matrix} \underset{ε \to 0}{lim sup} \int_{U_{ε}^{m}} {| A_{ε} \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ A_{ε} \nabla w_{ε}^{i} |}^{2} φ d x ⩽ C \int_{Ω} | σ - A \nabla ϕ | | \nabla T_{m} (u - ϕ) | φ d x . \end{matrix}$ Taking into account that $\begin{matrix} χ_{U_{ε}^{m}} \to χ_{{| u - ϕ |} ⩽ m}} a.e. in Ω, \end{matrix}$ for every $m \in N$ outside a countable set, (3.22), (3.29), and reasoning by semicontinuity, we get $\begin{matrix} \int_{{| u - ϕ | < m}} | σ - A \nabla ϕ |^{2} φ d x ⩽ C \int_{Ω} | σ - A \nabla ϕ | | \nabla T_{m} (u - ϕ) | φ d x, \forall m > 0 . \end{matrix}$ By density, this is true for every $φ \in L^{\infty} (Ω)$ , $φ ⩾ 0$ in Ω, which shows that $\begin{array}{l} | σ - A \nabla ϕ |^{2} χ_{{| u - ϕ | ⩽ m}} ⩽ C | σ - A \nabla ϕ | | \nabla T_{m} (u - ϕ) | \\ a.e. in Ω, \forall ϕ \in H^{1} (Ω) \cap L^{\infty} (Ω), \forall m > 0 . \end{array}$ Thus, $σ = A \nabla u$ a.e. in Ω, which combined with (3.23) implies that the first equation in (2.27) is satisfied.

Step 6. To finish the proof let us now show the corrector result (2.29). For this purpose, we return to (3.32), which taking into account that $σ = A \nabla u$ , (2.1) and $A \in L^{\infty} {(Ω)}^{N \times N}$ implies $\begin{matrix} (3.33) & \underset{ε \to 0}{lim sup} \int_{U_{ε}^{m}} {| \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i} |}^{2} φ d x ⩽ C \int_{{| u - ϕ | ⩽ m}} {| \nabla (u - ϕ) |}^{2} φ d x, \end{matrix}$ for every $ϕ \in C^{\infty} (Ω)$ , and every $φ \in C_{c}^{\infty} (Ω)$ , $φ ⩾ 0$ in Ω.

Now, for $h \in N$ fixed, we use that for every $φ \in C_{c}^{\infty} (Ω)$ , with $0 ⩽ φ ⩽ 1$ a.e. in Ω, we have $\begin{array}{l} \int_{{| u_{ε} | ⩽ h}} | \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i} | d x \\ ⩽ \int_{{| u_{ε} | ⩽ h}} | \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i} | (1 - φ) d x \\ + \int_{{| u_{ε} | ⩽ h} ∖ U_{ε}^{m}} | \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i} | φ d x + \int_{U_{ε}^{m}} | \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i} | φ d x . \end{array}$ Passing to the limit when ε tends to zero, this proves $\begin{array}{l} \underset{ε \to 0}{lim sup} \int_{{| u_{ε} | ⩽ h}} | \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i} | d x \\ ⩽ Λ {(\int_{{| u_{ε} | ⩽ h}} {(1 - φ)}^{2} d x)}^{\frac{1}{2}} + Λ {| {| u | ⩽ h} ∖ {| u - ϕ | ⩽ m} |}^{\frac{1}{2}} \\ + C {(\int_{{| u | ⩽ h}} {| \nabla (u - ϕ) |}^{2} d x)}^{\frac{1}{2}} \end{array}$ where Λ is a constant depending on ${sup}_{ε > 0} ‖ T_{h} (u_{ε}) ‖_{H_{0}^{1} (Ω)}$ , ${sup}_{ε > 0} ‖ w_{ε}^{i} ‖_{H^{1} (Ω)}$ , $1 ⩽ i ⩽ N$ , and $‖ \nabla ϕ ‖_{L^{\infty} {(Ω)}^{N}}$ .

By densitiy, we can take φ in $L^{\infty} (Ω)$ and then $φ = χ_{{| u | ⩽ h}}$ . Taking later m tending to infinity, we get $\begin{matrix} \underset{ε \to 0}{lim sup} \int_{{| u_{ε} | ⩽ h}} | \nabla u_{ε} - \sum_{i = 1}^{N} \partial_{i} ϕ \nabla w_{ε}^{i} | d x ⩽ C {(\int_{{| u | ⩽ h}} {| \nabla (u - ϕ) |}^{2} φ d x)}^{\frac{1}{2}} \end{matrix}$ for every $ϕ \in C^{\infty} (\overline{Ω})$ . Replacing ϕ by a sequence which converges to $T_{h} (u)$ in $H^{1} (Ω)$ , we conclude (2.29). □

Footnotes

Acknowledgements

This paper has been partially supported by the Ministerio de Ciencia e Innovación PID2020-116809GB-I00 of Spain.

This work has been carried out from an invitation from the Department of Mathematics of the University Roma La Sapienza. The author is deeply grateful.

References

Bénilan,

Boccardo,

Gallouët,

Gariepy,

Pierre and

J.L.

Vázquez, An

L^{1}

-theory of existence and uniqueness of solutions on nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci 22(2) (1995), 241–273.

Boccardo, Dirichlet problems with singular convection terms and applications, J. Differential Equations 258(7) (2015), 2290–2314. doi:10.1016/j.jde.2014.12.009.

Boccardo and

Casado-Díaz, H-convergence of singular solutions of some Dirichlet problems with terms of order one, Asymptot. Anal. 64(3–4) (2009), 239–249. doi:10.3233/ASY-2009-0943.

Boccardo and

Orsina, Regularizing effect of the lower order terms in some elliptic problems: Old and new, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29(2) (2018), 387–399. doi:10.4171/RLM/812.

Boccardo and

Orsina, Sublinear elliptic system with a convection term, Comm. Partial Differential Equations 45(7) (2020), 690–713. doi:10.1080/03605302.2020.1712417.

Briane, Homogenization with an oscillating drift: From

L^{2}

-bounded to unbounded drifts, 2d compactness results and 3d nonlocal effect, Ann. Mate. Pura Appl. 192(5) (2013), 853–878. doi:10.1007/s10231-012-0249-y.

Briane and

Casado-Díaz, A class of second-order linear elliptic equations with drift: Renormalized solutions, uniqueness and homogenization, Potential Anal. 43(3) (2015), 399–413. doi:10.1007/s11118-015-9478-1.

Briane and

Casado-Díaz, Increase of mass and nonlocal effects in the homogenization of magneto-elastodynamics problems, Calc. Var. 60 (2021), 163. doi:10.1007/s00526-021-02027-0.

Briane and

Gérard, A drift homogenization problem revisited, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 11(5) (2012), 1–39.

10.

Dal Maso,

Murat,

Orsina and

Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa 28 (1999), 741–808.

11.

Fabrie and

Gallouët, Modeling wells in porous media flow, Math. Models Methods Appl. Sci. 10(5) (2000), 673–709. doi:10.1142/S0218202500000367.

12.

Hillen and

K.J.

Painter, A user’s guide to PDE models for chemotaxis, J. Math. Biol. 58(1–2) (2009), 183–217. doi:10.1007/s00285-008-0201-3.

13.

N.G.

Meyers, An

L^{p}

-estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3(17) (1963), 189–206.

14.

Murat, Homogenization of renormalized solutions of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 8(3–4) (1991), 309–332. doi:10.1016/S0294-1449(16)30266-9.

15.

Murat, H-convergence. Séminaire d’Analyse Fonctionnelle et Numérique, 1977–78, Université d’Alger, multicopied, 34 pp. English translation: F. Murat and L. Tartar. H-convergence. In Topics in the Mathematical Modelling of Composite Materials (L. Cherkaev, R.V. Kohn, eds.) Progr. Nonlin. Differ. Equ. Appl. Vol. 31. Birkaüser, Boston (1998) 21–43.

16.

Spagnolo, Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 22 (1968), 571–597.

17.

Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier 15 (1965), 189–258. doi:10.5802/aif.204.

18.

Tartar, Homogénéisation en hydrodynamique, in: Singular Perturbation and Boundary Layer Theory,

C.-M.

Brauner,

Gay and

Mathieu, eds, Lecture Notes Math., Vol. 597, Springer, Berlin, Heidelberg, 1977, pp. 474–481. doi:10.1007/BFb0086103.

19.

Tartar, The General Theory of Homogenization: A Personalized Introduction, Lect. Notes Unione Matematica Italiana., Springer-Verlag, Berlin, Heidelberg, 2009.