Abstract
In this paper we deal with asymptotic behaviour of renormalized solutions \(u_{n}\) to the nonlinear parabolic problems whose model is
where \(\Omega \) is a bounded open set of \(\mathbb {R}^{N}\), \(N\ge 1\), \(T>0\) and \(u_{0}^{n}\in C^{\infty }_{0}(\Omega )\) that approaches \(u_{0}\) in \(L^{1}(\Omega )\). Moreover \((\mu _{n})_{n\in \mathbb {N}}\) is a sequence of Radon measures with bounded variation in Q which converges to \(\mu \) in the narrow topology of measures. The main result states that, under the assumption of G-convergence of the operators \(A_{n}(v)=-\text {div}(a_{n}(t,x,\nabla v_{n}))\), defined for \(v_{n}\in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) for \(p>1\), to the operator \(A_{0}(v)=-\text {div}(a_{0}(t,x,\nabla v))\) and up to subsequences, \((u_{n})\) converges a.e. in Q to the renormalized solution u of the problem
The proposed renormalized formulation differs from the usual one by the fact that truncated function \(T_{k}(u_{n})\) (which depend on the solutions) are used in place of the solutions \(u_{n}\). We prove existence of such a limit-solution and we discuss its main properties in connection with G-convergence, we finally show the relationship between the new approach and the previous ones and we extend this result using capacitary estimates and auxiliary test functions.
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1 Introduction
We are interested in the asymptotic behaviour of the solutions \(u_{n}\) to the (homogeneous) parabolic problems
where \(\Omega \) is a bounded open subset of \(\mathbb {R}^{N}\), \(N\ge 2\), \((u_{0}^{n})_{n\in \mathbb {N}}\) is a sequence of smooth functions that approaches \(u_{0}\) in \(L^{1}(\Omega )\), \((\mu _{n})_{n\in \mathbb {N}}\) is a sequence of Radon measures with bounded total variation on \(Q=(0,T)\times \Omega \) which converges to \(\mu \) in the narrow topology of measures, and \((a_{n}(t,x,\zeta ))_{n\in \mathbb {N}}\) are Carathéodory functions such that the corresponding operators \(A_{n}:L^{p}(0,T,W^{1,p}_{0}(\Omega ))\mapsto L^{p'}(0,T;W^{-1,p'}(\Omega ))\), defined by \(A_{n}(v_{n})=-\text {div }(a_{n}(t,x,\nabla v_{n}))\), turn out to be monotone, continuous and coercive between the Sobolev space \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\), \(p>1\), and its dual space \(L^{p'}(0,T,W^{-1,p'}(\Omega ))\), \(\frac{1}{p}+\frac{1}{p'}=1\). Under these assumptions, for every \(F_{n}\in L^{p'}(Q)\) and \(u_{0}\in L^{2}(\Omega )\) there exists a unique variational solution \(v_{n}\) to the problem
that is a unique function \(v_{n}\in W\cap C(0,T;L^{2}(\Omega ))\) in the weak sense
where \(\langle \cdot ,\cdot \rangle \) denotes the usual duality pairing between the spaces \(W^{1,p}_{0}(\Omega )\cap L^{2}(\Omega )\) and \((W^{-1,p'}(\Omega )\cap L^{2}(\Omega ))'\) (see [46] for the case \(p\ge 2\) and [44] for \(1<p<2\)). Let emphasize that variational solutions are solutions to problems with \(L^{p'}(Q)\)-data. This kind of problems has been largely studied in different context, but here the obtained results are related to the theory of homogenization, which is the study of the asymptotic behaviour for solutions of (1.1) corresponding to \(a_{n}(t,x,\zeta )=a(nt,nx,\zeta )\), \(n\in \mathbb {N}\), with suitable periodicity assumptions on \(a(\cdot ,\cdot ,\zeta )\). If there exists \(A_{0}(v)=-\text {div}(a_{0}(t,x,\nabla v)): L^{p}(0,T;W^{1,p}_{0}(\Omega ))\mapsto L^{p'}(0,T;W^{-1,p'}(\Omega ))\) such that for every \(F\in L^{p'}(Q)\), the solutions \(v_{n}\) of (1.1) converge weakly in \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) to the variational solution v of the problem
and the momenta \(a_{n}(t,x,\nabla v_{n})\) converge to \(a_{0}(t,x,\nabla v)\) weakly in \((L^{p'}(Q))^{N}\), the sequence \(A_{n}\) is said to be G-converging (or H-converging) to \(A_{0}\). Hence the sequence of operators \((A_{n})\)G-converges to \(A_{0}\) if the asymptotic behaviour of solutions corresponding to \(A_{n}\) is described by the problem corresponding to \(A_{0}\). In the case of the theory of Homogenization: the operator \(A_{0}\) represents the macroscopic model associated to the microscopic structures described, at different scale, by \(A_{n}\). Recall that the notion of G-convergence was introduced in Spagnolo [73] for parabolic operators (the name comes from the fact that G-convergence is defined in terms of Green’s operators), the extension to the elliptic case is defined in [72], especially to the second order symmetric linear elliptic operators in [71] (See also [55, 68, 69] for problems with lower order terms and [50, 70, 78] for non-symmetric case where the last two authors use the name H-convergence, H stands for Homogenization). For the case of elliptic operators with arbitrary order we refer to [54, 79], for the parabolic case we refer to Colombini and Spagnolo [19, 20] and Spagnolo [74], while the arbitrary order case is studied in [80] and for a class of non-divergence parabolic operators in [81]. The properties of G-convergence for some classes of quasi-linear elliptic operators with linear principal part are studied in [5, 8, 10, 58] and also [40] for the study of conditions under which the weak convergence of coefficients implies the G-convergence of the corresponding operators. The case of quasi-linear monotone operators in divergence form was studied by Tartar (unpublished notes, 1981), by Pankov [56], Del Vecchio [28], and Francu [32] under some equi-continuity assumptions, by Chiado Piat et al. [18] and Defranceschi [27] without any continuity conditions. The relationship between G-convergence of quasi-linear elliptic monotone operators and \(\Gamma \)-convergence of the corresponding convex functionals is studied in [26], the case of degenerate monotone operators in divergence form is considered in [25]. A discrete notion of G-convergence for finite difference equations can be found in [41], we refer to [39, 42, 43] for the case of quasi-linear parabolic operators and [21, 75] for hyperbolic case (see also [14, 22, 38, 57] for more references). As a consequence of stability properties proved in the context of Dirichlet problems with elliptic operators and measure data [23, 48, 49], we drive such new results for parabolic problems with measures which depend on time and the extension to more general spaces seems to be always possible. We consider, as starting point, a sequence \((A_{n})\) of operators in divergence form G-converging to an operator \(A_{0}\) of the same form. Hence \(A_{0}\) is a good model for the asymptotic behaviour of the variational solutions of (1.1). The main point is to study the possibility to describe the asymptotic behaviour of solutions corresponding to the operators \(A_{n}\) in the case where \(\mu \) lies in the space of measures, some partial results can be found in [67] when \(\mu \) lies in \(L^{1}(Q)\) and in [1, 61] in the case of “uniform” convergence of \(a_{n}(t,x,\zeta )\) (see for instance [62] for nonexistence results (concentration phenomena) and [2] for blowing-up problems). In particular if \(0<\alpha <\beta \) and \(\mathcal {M}(\alpha ,\beta )\) is the set of all matrices \(A(t,x)\in (L^{\infty }(Q))^{N\times N}\) such that
and if \((A_{n})\in \mathcal {M}(\alpha ,\beta )\)G-converges to \(A_{0}\in \mathcal {M}(\alpha ,\beta )\), then the above assumptions implies that the adjoint matrices \((\overline{A}_{n})\)G-converges to the adjoint \(\overline{A}_{0}\) of \(A_{0}\). Letting \(\mu \) be a fixed measure with bounded variation in Q, we consider the solutions \(u_{n}\) of problems
Regardless of the assumptions on \(A_{n}\), compactness results on the solutions of problems (1.2) plays a crucial role in the existence and uniqueness of solutions \((u_{n})\) (see [9, 66, 76]), \(u_{n}\) is a solution of (1.2) if \(u_{n}\in L^{q}(0,T;W^{1,q}_{0}(\Omega ))\cap L^{\infty }(0,T;L^{1}(\Omega ))\) for every \(q<p-\frac{N}{N+1}\) and, for every \(f_{n}\in L^{\infty }(Q)\), \(u_{n}\) satisfies
where \(v_{n}\) is the variational solution to
On the other hand, solution \(u_{n}\) of (1.2) is unique and that there exists \(C>0\), depending only on N, \(\alpha \) and \(\beta \), such that \(\Vert u_{n}\Vert _{L^{q}(0,T;W^{1,q}_{0}(\Omega ))}\le C\) (see [60, 66]). Precisely, we can extract a subsequence \((u_{n})\) which converges weakly in \(L^{q}(0,T;W^{1,q}_{0}(\Omega ))\) to a function u. In particular, \((u_{n})\) converges to u strongly in \(L^{1}(Q)\), and then
for every \(f\in L^{\infty }(Q)\). As a consequence of G-convergence hypothesis on the operators, the solutions \(v_{n}\) converge weakly in \(L^{2}(0,T;H^{1}_{0}(\Omega ))\) to the solution v of problem
As a consequence of classical regularity results, \((v_{n})\) turns out to be a sequence of equi-Hölder continuous functions and hence the sequence converges to v uniformly in Q. We then have
which, together with (1.3)–(1.4), implies that u is solution of problem
and the whole sequence \((u_{n})\) converges to u. Losely speaking, the method of G-convergence of linear operators with measures allows to describe a similar model valid for variational solutions. It is worth noting that, in this paper, such a construction can be achieved (at least with the same technique) as far as nonlinear parabolic operators are concerned. Note that in this case we miss two important properties: first, the fact that the formulation in terms of duality (1.3) translates the problem of the passage to the limit for solutions corresponding to measure data in a problem of convergence of solutions corresponding to regular data (solved by the assumption of G-convergence) (see [60, 76]), Second, the property that in the linear case the solution corresponding to measure data is unique, while for nonlinear equations the uniqueness of the solution for general measure data is still an open problem [61, 63], we would like to emphasize that due to the first difficulty we are led to choose to set our results in the framework of the so-called renormalized solution given in [61] in the context of parabolic problems
Let us recall that the main property of renormalized solutions \(u_{n}\) of (1.5) is that all truncations \(T_{k}(u_{n})\) are variational solutions of the boundary value problems (adapted to its truncations) defined by
where \(\mu _{k,n}\) are suitable regular measures (more precisely \(\mu _{k,n}\) does not charge sets of zero parabolic p-capacity) which converges to \(\mu \) as k goes to \(+\infty \). Our results are actually in the same spirit as those in [48] which concern the elliptic equation
The paper is planned in the following way, in Sect. 2 we will precise the notion of capacity and some basic properties of measures and we will state the main result using assumptions on Leray–Lions operators and the definitions of renormalized solutions, whose proof, which is rather technical, is left to Sect. 3, where the strategy is to pass to the limit as \(n\rightarrow +\infty \) in (1.6) for every \(k>0\) fixed, instead of trying to pass to the limit in the original problems. This approach has the advantage of attacking the problem from a variational point of view. Nevertheless, the passage to the limit in (1.6) cannot be performed directly using the hypothesis of G-convergence of the operators, due to the presence of varying right-hand sides, that converge only in a very weak sense. First we prove some a priori estimates on the elements \(u_{n}\), \(T_{k}(u_{n})\), \(a_{n}(t,x,\nabla u_{n})\), \(a_{n}(t,x,\nabla T_{k}(u_{n}))\) and \(\mu _{k,n}\) in order to obtain a limit equation where information about operators and data are lost (Sect. 4). As a consequence, we reconstruct the datum in Sect. 5, and following the approach of Minty’s Lemma, we reconstruct the operator in Sect. 6. Because of the lack of uniqueness result, we obtain that for every fixed \(\mu \) and for every choice of a sequence of renormalized solutions to problem (1.5) it is possible to extract a subsequence (possibly depending on \(\mu \)) which converges to a renormalized solution of the problem corresponding to the G-limit \(A_{0}\) and with datum \(\mu \).
2 Preliminaries and general results
Let \(\Omega \subseteq \mathbb {R}^{N}\) be a bounded open set, \(N\ge 2\), and let p and \(p'\) be two real numbers with \(1<p\le N\) and \(\frac{1}{p}+\frac{1}{p'}=1\). Let us fix some notations. Henceforward, we will consider, respectively, \(|\zeta |\) and \(\zeta \cdot \zeta '\) the Euclidean norm of a vector \(\zeta \in \mathbb {R}^{N}\) and the scalar product between \(\zeta \) and \(\zeta '\in \mathbb {R}^{N}\). For formally, a certain property holds almost everywhere (or a.e.) if it holds for all cases except for a certain subset which is very small. Frequently it will be convenient to describe situations that hold except on sets of zero measure. So by convention, a property is said to hold \(\mu \)-almost everywhere \((\mu \text {-a.e.})\) if the set of points on which it doesn’t hold has \(\mu \)-measure zero, similarly, this notations can be used in the case of convergence. Moreover, in what follows, \(\omega \) will indicate any quantity that vanishes as the parameters in its argument go to their (obvious, if not explicitly stressed) limit point with the same order in which they appear.
The function \(T_{k}(s)\)
2.1 Capacity
For every Borel set \(B\subseteq Q\), its p-capacity \(\text {cap}_{p}(B,Q)\) with respect to Q is defined by (see [37, 65])
where the infimum is taken over all the functions \(u\in W\) such that \(u\ge 1\) a.e. in a neighborhood of B. We say that a property \(\mathcal {P}(t,x)\) holds \(\text {cap}_{p}\)-quasi everywhere if \(\mathcal {P}(t,x)\) holds for every (t, x) outside a subset of Q of zero p-capacity. A function u defined on Q is said to be \(\text {cap}_{p}\)-quasi continuous if for every \(\epsilon >0\) there exists \(B_{\epsilon }\subseteq Q\) with \(\text {cap}_{p}(B_{\epsilon },Q)<\epsilon \) such that the restriction of u to \(Q\backslash B\) is continuous. It is well known that every function in \(W_{2}\) defined by
has a unique \(\text {cap}_{p}\)-quasi continuous representative \(\tilde{u}\), whose values are defined (and finite) \(\text {cap}_{p}\)-quasi everywhere in Q. In what follows we always identify a function \(u\in W_{2}\) with its \(\text {cap}_{p}\)-quasi continuous representative \(\tilde{u}\). A set \(E\subseteq Q\) is said to be \(\text {cap}_{p}\)-quasi open if for every \(\epsilon >0\) there exists an open set \(U_{\epsilon }\) such that \(E\subseteq U_{\epsilon }\subseteq Q\) and \(\text {cap}_{p}(U\backslash E,Q)\le \epsilon \). It can be easily seen that, if u is a \(\text {cap}_{p}\)-quasi continuous function, then for every \(k\in \mathbb {R}\) the sets \(\lbrace u>k\rbrace =\lbrace (t,x)\in Q:u(t,x)>k\rbrace \) and \(\lbrace u<k\rbrace =\lbrace (t,x)\in Q:u(t,x)<k\rbrace \) are \(\text {cap}_{p}\)-quasi open. The characteristic function of a \(\text {cap}_{p}\)-quasi open set can be approximated by a monotonic sequence of functions in the energy space \(W_{2}\), as stated in the following lemma (see [31, Theorem 2.11, Lemma 2.20]).
Lemma 2.1
For every \(\text {cap}_{p}\)-quasi open set \(E\subseteq Q\), there exists an increasing sequence \((w_{n})\) of nonnegative functions in W which converges to \(\chi _{E}\)\(\text {cap}_{p}\)-quasi everywhere in Q.
2.2 Truncations
For every \(k>0\), we define the truncation function \(T_{k}:\mathbb {R}\mapsto \mathbb {R}\) (see Fig. 1) by \(T_{k}(s):\text {max}(-k,\text {min}(k,s))\). Let us consider the space \(\mathcal {T}^{1,p}_{0}(Q)\) of all functions \(u:Q\rightarrow \overline{\mathbb {R}}\) which are measurable and finite a.e. in Q, and such that \(T_{k}(u)\) belongs to \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) for every \(k>0\). It is easy to see that every function \(u\in \mathcal {T}^{1,p}_{0}(Q)\) has a \(\text {cap}_{p}\)-quasi continuous representative, that will always be identified with u. Moreover, for every \(u\in \mathcal {T}^{1,p}_{0}(Q)\), there exists a measurable function \(v:Q\mapsto \mathbb {R}^{N}\) such that \(\nabla T_{k}(u)=v\chi _{\lbrace |u|\le k\rbrace }\) a.e. in Q, for every \(k>0\), and v is unique up to almost everywhere equivalence [4, Lemma 2.1]. Hence it is possible to define a generalized gradient \(\nabla u\) of \(u\in \mathcal {T}^{1,p}_{0}(Q)\), setting \(\nabla u=v\). If \(u\in L^{1}(0,T;W^{1,1}(\Omega ))\), the gradient coincides with the usual one, while for \(u\in L^{1}(0,T;L^{1}_{\text {loc}}(\Omega ))\), it may differ from the distributional gradient of u.
2.3 Measures
Let us denote with \(\mathcal {M}_{b}(Q)\) the set of all Radon measures on Q with bounded total variation and \(C_{b}(Q)\) the space of all bounded, continuous functions on Q, so that \(\int _{Q}\varphi \mathrm{{d}}\mu \) is defined for \(\varphi \in C_{b}(Q)\) and \(\mu \in \mathcal {M}_{b}(Q)\) where the positive, the negative and the total variation parts of a measure \(\mu \) in \(\mathcal {M}_{b}(Q)\) are denoted by \(\mu ^{+}\), \(\mu ^{-}\) and \(|\mu |\), respectively. We recall that for a measure \(\mu \) in \(\mathcal {M}_{b}(Q)\) and a Borel set \(E\subseteq Q\), the measure 
is defined by 
for any Borel set \(B\subseteq Q\). Analogous, we define \(\mathcal {M}_{0}(Q)\) as the set of all measures \(\mu \) in \(\mathcal {M}_{b}(Q)\) with bounded variation over Q that does not charge the sets of zero parabolic p-capacity that is if \(\mu \in \mathcal {M}_{0}(Q)\) then \(\mu (B)=0\) for every Borel set \(B\subseteq Q\) such that \(\text {cap}_{p}(B,Q)=0\), while \(\mathcal {M}_{s}(Q)\) will be the set of all measures \(\mu \) in \(\mathcal {M}_{b}(Q)\) for which there exists a Borel set \(E\subset Q\), with \(\text {cap}_{p}(E,Q)=0\), such that 
.
Remark 2.2
A measure \(\mu _{0}\in \mathcal {M}_{0}(Q)\) if and only if for every \(\epsilon >0\), there exists \(\delta >0\) such that \(\mu _{0}(B)<\epsilon \) for every Borel set \(B\subseteq Q\) with \(\text {cap}_{p}(B,Q)<\delta \).
Proposition 2.3
If \(\mu _{0}\in \mathcal {M}_{0}(Q)\) and if v is a function in \(W_{2}\). Then v is measurable with respect to \(\mu _{0}\). If v further belongs to \(L^{\infty }(Q)\), then v belongs to \(L^{\infty }(Q,\mu _{0})\) and \(\Vert v\Vert _{L^{\infty }(Q,\mu _{0})}=\Vert v\Vert _{L^{\infty }(Q)}\).
Proof
See [59, corollary 4.9]. \(\square \)
Thanks to this result and to the dominated convergence theorem, we derive the following limit
Corollary 2.4
If \(\mu _{0}\in \mathcal {M}_{0}(Q)\) and \((v_{n})\in W_{2}\cap L^{\infty }(Q)\), bounded in \(L^{\infty }(Q)\), which converges to a function v\(\text {cap}_{p}\)-quasi everywhere. Then \((v_{n})\) converges to v\(\mu _{0}\)-almost everywhere, and
Remark 2.5
Let (\(\rho _{n})\) be a sequence of \(L^{1}(Q)\)-functions converging to \(\rho \) weakly in \(L^{1}(Q)\). Let \((\Theta _{n})\) be a sequence of functions belonging to \(L^{\infty }(Q)\), bounded in the same space, and converging a.e. in Q to a function \(\Theta \). Then, according to the Egorov theorem, we have
this result will be often used in what follows.
On the other hand, if \((v_{n})\) is a sequence of functions in \(W_{2}\) which converges weakly to v, then for every \(\mu _{0}\in \mathcal {M}_{0}(Q)\) the sequence \((v_{n})\) converges to v in \(\mu _{0}\)-measure (see [16]). Before passing to the convergence results, let us state an interesting result about the decomposition of measures in \(\mathcal {M}_{0}(Q)\).
Lemma 2.6
If \(\mu _{0}\in \mathcal {M}_{0}(Q)\) then, there exist a decomposition (f, G, g) of \(\mu _{0}\) such that \(\mu _{0}=f-{\text {div}}(G)+g_{t}\) where \(f\in L^{1}(Q)\), \(G\in (L^{p'}(Q))^{N}\) and \(g\in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\cap L^{2}(\Omega )\). Moreover
for every \(v\in W\cap L^{\infty }(Q)\).
Proof
See [31, Theorem 2.1]. \(\square \)
The standard argument of Lemma 2.6 plays a key role in the proof of the following convergence result.
Lemma 2.7
If \(\mu _{0}\in \mathcal {M}_{0}(Q)\) and \((v_{n})\) is a sequence of functions in \(W_{2}\cap L^{\infty }(Q)\) which converges to a function \(v\in W_{2}\cap L^{\infty }(Q)\) weakly in \(W_{2}\) . Assume that \(\Vert v_{n}\Vert _{L^{\infty }(Q)}\le C\) for every \(n\in \mathbb {N}\). Then \((v_{n})\) converges to v strongly in \(L^{2}(Q,\mu _{0})\).
Let us state a general decomposition result of measures in \(\mu \in \mathcal {M}_{b}(Q)\) and used several times in the next.
Theorem 2.8
Let \(\mu \in \mathcal {M}_{b}(Q)\), then there exists a unique decomposition \((\mu _{0},\mu _{s})\) such that \(\mu =\mu _{0}+\mu _{s}\), \(\mu _{0}\in \mathcal {M}_{0}(Q)\) and \(\mu _{s}\in \mathcal {M}_{s}(Q)\).
Proof
See [33, Lemma 2.1]. \(\square \)
Recall that a sequence \((\mu _{n})\) of measures in \(\mathcal {M}_{b}(Q)\) converges to a measure \(\mu \) in \(\mathcal {M}_{b}(Q)\) in the narrow topology of measures if
for every \(\varphi \in C_{b}(Q)\). If (2.2) holds for all continuous functions \(\varphi \) with compact support in Q (i.e. \(\varphi \in C_{c}(Q)\)), then it coincides with the usual weak-* convergence in \(\mathcal {M}_{b}(Q)\).
Remark 2.9
Recall also that a sequence of nonnegative measures \((\mu _{n})\) converges to \(\mu \) in the narrow topology if and only if it converges to \(\mu \) in the weak-* topology, and the masses \((\mu _{n}(Q))\) converges to \(\mu (Q)\). Then, for nonnegative measures, the narrow convergence is equivalent to the convergence in (2.2) for every \(\varphi \in C^{\infty }(\overline{Q})\).
2.4 The operator
A function \(a:Q\times \mathbb {R}^{N}\rightarrow \mathbb {R}^{N}\) is said to be a Carathéodory function if \(a(\cdot ,\cdot ,\zeta )\) is measurable on Q for every \(\zeta \in \mathbb {R}^{N}\) and \(a(t,x,\cdot )\) is continuous on \(\mathbb {R}^{N}\) for almost every (t, x) in Q. Fixed two constants \(c_{0}\), \(c_{1}>0\) and a nonnegative function \(b_{0}\in L^{s}(Q)\), \(s>\frac{N}{p}\), we say that \(a:Q\times \mathbb {R}^{N}\rightarrow \mathbb {R}^{N}\) satisfies hypothesis \(H(c_{0},c_{1},b_{0})\) if for almost every \((t,x)\in Q\) the following assumptions hold
Remark 2.10
We observe that if \(a(t,x,\zeta )\) is a Carathéodory function satisfying (2.3), then \(a(t,x,0)=0\) for a.e. (t, x) in Q.
Thanks to the assumptions (2.3)–(2.5), the differential operator \(u\mapsto -\text {div}(a(t,x,\nabla u))\) turns out to be a coercive and monotone operator acting from the space \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) into its dual space \(L^{p'}(0,T;W^{-1,p'}(\Omega ))\). It is well known, by the standard theory of monotone operators (see for instance [45, 46]), if \(F\in L^{p'}(Q)\), then there exists a unique variational solution v of the problem
in the sense that v belongs to \(W\cap C(0,T;L^{2}(\Omega ))\) and
for all \(\varphi \in W\text { such that }\varphi (T)=0\), \(\langle \cdot ,\cdot \rangle \) denotes the duality pairing between \(W^{-1,p'}(\Omega )\) and \(W^{1,p}_{0}(\Omega )\).
Remark 2.11
We also recall that, since \(\partial \Omega \) is smooth and a satisfies assumptions \(H(c_{0},c_{1},b_{0})\) with \(b_{0}\in L^{s}(Q)\) with \(s>\frac{N}{p}\), for every \(F=-\text {div}(G_{0})\), \(G_{0}\in (L^{\infty }(Q))^{N}\) the solution v of (2.6) is a Hölder continuous function. Moreover, there exists \(C>0\), depending only on p, \(c_{0}\), \(c_{1}\), \(b_{0}\) and \(\mathcal {L}^{N}(Q)\), such that \(\Vert v\Vert _{C^{0,\alpha }(Q)}\le C\Vert G_{0}\Vert _{(L^{\infty }(Q))^{N}}\) (see [35, 36, 45]).
Let us define a \(\mathcal {M}_{0}\)-version of Minty’s Lemma (an elliptic version of this Lemma can be found in [11, Lemma 7] and different type of Minty’s Lemma are established in [13] for parabolic problems with p-growth). Our Minty type Lemma can be proved as the elliptic case and reads as follows
Lemma 2.12
Let a be Carathéodory function satisfying \(H(c_{0}, c_{1},b_{0})\), let \(\gamma _{0}\) be measure in \(\mathcal {M}_{0}(Q)\) and v be a function in \(\mathcal {T}^{1,p}_{0}(Q)\). Then v is such that
for every \(w\in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\cap L^{\infty }(Q)\cap C(0,T;L^{1}(\Omega ))\) if and only if v satisfies
for every \(w\in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\cap L^{\infty }(Q)\cap C(0,T;L^{1}(\Omega ))\) with \(w_{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega ))\).
In order to better specify the definition of G-convergence (also called Homogenization, Fig. 2), recall that the present ideas of this paper differs from the several papers cited above. In fact, in other works, especially parabolic problems, the G-convergence is related to finite energy solutions. But here, this concept fits with the renormalized framework, it deals with the homogenization of the renormalized formulation in the case where the sequence of momenta \((a_{n})\) considered, bounded, in \((L^{p'}(Q))^{N}\) and satisfies (2.3)–(2.5) for some fixed \(\alpha ,\beta >0\). It consists in proving that if \(u_{0}^{n}\in L^{1}(\Omega )\), \((a_{n})\)G-converges to \(a_{0}\) (see the definition of this convergence in Definition 2.13 bellow) and that the sequence of measures \((\mu _{n})_{n\in \mathbb {N}}\) tightly converges to \(\mu \) (i.e., in the narrow topology of measures), a subsequence of the sequence of the solutions of the renormalized equation relative to \(a_{n}\) converges to a solution of the renormalized equation relative to \(a_{0}\), this is in order to see that the notion of renormalized solution is thus robust in the sense that it is stable under the G-convergence of the momenta, which is the “weakest possible” convergence for the corresponding operators. It is also worth noticing that the proof that we will present in Sect. 6 to prove this homogenization result of renormalized solutions is closed to the proof used in [48] (see also [52]) to obtain the asymptotic behaviour of renormalized solutions and to illustrate the robustness of the method in elliptic case. The robustness of both the notion of renormalized solution and of the method of proof (using truncate functions) is emphasized by the stability result of renormalized solutions with respect to variations of the right-hand side which given in the following classical result: Consider a sequence of weak solutions \(u_{n}\) relative to some operators \((a_{n})\) and to a sequence of right-hand sides \(F_{n}\) which converges weakly in \(L^{p'}(0,T;W^{-1,p'}(\Omega ))\) to F (see 2.10) and under a special assumptions of equi-integrability of \(F_{n}\), a subsequence of the sequences \(u_{n}\) is proved to converge weakly in \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) to a solution of the weak equation relative to the right-hand side F. The above main result was generalized by Malusa and Orsina [48] under Leray–Lions and G-convergence assumptions in the case of singular measures, this argument, however, quite simply extends to measures converging weak-* was observed in [47] in order to prove the asymptotic behavior of Stampacchia solutions and under weak-* convergence of the data (fixed measures) in the theory of “cheap” control. Moreover, the asymptotic behaviour (G-convergence, \(H-\)convergence, numerical approximation) of the solutions of Dirichlet problems in \(L^{1}(\Omega )\) can be found in [6, 7, 12, 17, 24, 53], All these homogenization elliptic problems are considered Dirichlet setting, for which the solutions is zero in the boundary. In a different setting, let us mention [15] where renormalized solutions and homogenization are mixed and the more recent work [34] where the authors study, with the help of renormalized solutions, the homogenization of a linear elliptic problem with \(L^{1}\)-data, Neumann boundary conditions and highly oscillating boundary. In addition, Ben Cheikh Ali in [3] studied the homogenization of a renormalized solution in perforated domains with a Neumann boundary condition on the boundary of the holes and the authors in [30] considered the homogenization of a class of quasilinear elliptic problems in a periodically perforated domain with \(L^{1}\)-data and nonlinear Robin conditions on the boundary of the holes. Observe that variational solutions corresponding to the data \(f\in L^{p'}(Q)\) are renormalized solutions corresponding to the measure \(\hbox {d}\mu =f \mathrm{{d}}x\mathrm{{d}}t\) and the narrowly convergence implies the \(*\)-weak convergence, that is, if \(\mu _{n}\) converges in the narrow topology of measures to \(\mu \) then \(\mu _{n}\) converges to \(\mu \)*-weakly in \(\mathcal {M}_{b}(Q)\) and \(\mu _{n}(Q)\) converges to \(\mu (Q)\). We now recall the definition of G-convergence related to parabolic operators.
Homogenization theory
Definition 2.13
Let \((a_{n})_{n\in \mathbb {N}}\) and \(a_{0}\) be Carathéodory functions satisfying \(H(c_{0},c_{1},b_{0})\), and let \(A_{n}(u)=-\text {div}(a_{n}(t,x,\nabla u))\), \(n\in \mathbb {N}\), and \(A_{0}(u)=-\text {div}(a_{0}(t,x,\nabla u))\) be the corresponding operators between the spaces \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) and \(L^{p'}(0,T;W^{-1,p'}(\Omega ))\). We say that \(A_{n}\)G-converges to \(A_{0}\) if for every \(F\in L^{p'}(0,T;W^{-1,p'}(\Omega ))\) and for every \(z\in L^{p}(0,T;W^{1,p}(\Omega ))\) the (variational) solutions \(v_{n}\) of problems
satisfy
where v is the (variational) solution of problem
2.5 Renormalized solutions
The main idea of renormalized solutions consists on multiplying the pointwise equation by a test function in dependence of u (any smooth function with compact support). Let \(W^{2,\infty }(\mathbb {R})\) is the set of all Lipschitz continuous functions \(h:\mathbb {R}\rightarrow \mathbb {R}\) whose derivative \(h'\) has compact support, i.e., every function \(h\in W^{2,\infty }(\mathbb {R})\) is constant outside the support of its derivative, so that we can define \(h(0)=0\), \(u_{g}=u-g\) where \(g_{t}\) is the time derivative part of \(\mu _{0}\) and \(\tilde{\mu }_{0}=\mu -g_{t}-\mu _{s}=f-\text {div}(G)\).
Definition 2.14
Let a be Carathéodory function satisfying \(H(c_{0},c_{1},b_{0})\), and let \(\mu \) be a measure in \(\mathcal {M}_{b}(Q)\), decomposed as \(\mu =\mu _{0}+\mu _{s}\), \(\mu _{0}\in \mathcal {M}_{0}(Q)\), \(\mu _{s}\in \mathcal {M}_{s}(Q)\). A function u is a renormalized solution of problem
if the following conditions hold
- (a)
\(u\in \mathcal {T}^{1,p}_{0}(Q)\);
- (b)
\(|\nabla u|^{p-1}\) belongs to \(L^{q}(Q)\) for every \(q<p-\frac{N}{N+1}\);
- (c)
For every \(h\in W^{2,\infty }(\mathbb {R})\) one has
$$\begin{aligned} \left\{ \begin{aligned}&-\int _{\Omega }h(u_{0})\varphi (0)\mathrm{{d}}x-\int _{0}^{T}\langle \varphi _{t},h(u_{g})\rangle \mathrm{{d}}t+\int _{Q}h'(u_{g})a(t,x,\nabla u)\cdot \nabla \varphi \mathrm{{d}}x \mathrm{{d}}t\\&+\int _{Q}h''(u_{g})a(t,x,\nabla u)\cdot \nabla u_{g}\varphi \mathrm{{d}}x \mathrm{{d}}t=\int _{Q}h'(u_{g})\varphi d\tilde{\mu }_{0}, \end{aligned} \right. \end{aligned}$$(2.13)for every \(\varphi \in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\cap L^{\infty }(Q)\), \(\varphi _{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega ))\), with \(\varphi (T,x)=0\), such that \(h'(u_{g})\varphi \in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\). Moreover, for every \(\psi \in C(\overline{Q})\) we have
$$\begin{aligned} \left\{ \begin{aligned}&\underset{n\rightarrow +\infty }{\text {lim}}\frac{1}{n}\int _{\lbrace n\le v< 2n\rbrace }a(t,x,\nabla u)\cdot \nabla u_{g}\psi \mathrm{{d}}x \mathrm{{d}}t=\int _{Q}\psi \mathrm{{d}}\mu _{s}^{+},\\&\underset{n\rightarrow +\infty }{\text {lim}}\frac{1}{n}\int _{\lbrace -2n< v\le n\rbrace }a(t,x,\nabla u)\cdot \nabla u_{g}\psi \mathrm{{d}}x \mathrm{{d}}t=\int _{Q}\psi \mathrm{{d}}\mu _{s}^{-}, \end{aligned} \right. \end{aligned}$$where \(\mu _{s}^{+}\) and \(\mu _{s}^{-}\) are respectively the positive and the negative parts of the singular part \(\mu _{s}\).
Let us point out that the existence of a renormalized solution of (2.12) is proved in [61] (see also [63] for another proof), the uniqueness of the solution for datum \(\mu =\mu _{0}\in \mathcal {M}_{0}(Q)\) is proved in [31, Sect 3] (see also [61, 67]), while the uniqueness of the renormalized solution for general \(\mu \in \mathcal {M}_{b}(Q)\) and initial \(u_{0}\in L^{1}(\Omega )\) is still open. The following equivalence is proved in [63, Sect 4] (using an analogous definition).
Theorem 2.15
Let u be a function satisfying \((a){-}(b)\) of Definition 2.14. Then u is a renormalized solution of problem (2.12) if and only if for every \(k>0\) there exist a sequence of non-negative measures \((\lambda _{k})\in \mathcal {M}_{b}(Q)\) such that
- (i)
\( \lambda _{k} \ \overrightarrow{{k \rightarrow \infty }} \ \mu _{s} \) in the narrow topology of measures;
- (ii)
the truncations \(T_{k}(u)\) satisfy
$$\begin{aligned} -\int _{Q}T_{k}(u)v_{t}\mathrm{{d}}x \mathrm{{d}}t+\int _{Q}a(t,x,\nabla T_{k}(u))\cdot \nabla v \mathrm{{d}}x \mathrm{{d}}t=\int _{Q}\tilde{v} \mathrm{{d}}\mu _{0}+\int _{Q}\tilde{v} d\lambda _{k}+\int _{\Omega }T_{k}(u_{0})v(0)\mathrm{{d}}x \end{aligned}$$(2.14)for every \(v\in W\cap L^{\infty }(Q)\) such that \(v(T)=0\) (with \(\tilde{v}\) being the unique \(\text {cap}_{p}\)-quasi continuous representative of v).
3 Statements of the main results
Let \((a_{n})_{n\in \mathbb {N}}\) and \(a_{0}\) be Carathéodory functions satisfying \(H(c_{0},c_{1},b_{0})\), and let \(A_{n}(u)=-\text {div }(a_{n}(t,x,\nabla u_{n}))\), \(A_{0}(u)=\text {div}(a_{0}(t,x,\nabla u))\) be the corresponding operators between \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) and \(L^{p'}(0,T;W^{-1,p'}(\Omega ))\).
Theorem 3.1
Let \(u_{0}^{n}\in L^{1}(\Omega )\) and assume that the sequence of operators \((A_{n})_{n\in \mathbb {N}}\)G-converge to \(A_{0}\) and that the sequence of measures \((\mu _{n})_{n\in \mathbb {N}}\) converges to \(\mu \) in the sense of (2.2). If \(u_{n}\) is a sequence of renormlized solutions of problem
Then, up to subsequences, \((u_{n})\) converges a.e. in Q to a function \(u\in \mathcal {T}^{1,p}_{0}(Q)\) renormalized solution of the problem
Moreover, we have, for every \(k>0\), the truncation functions \(T_{k}(u_{n})\) satisfy
It suffices to use the definition of renormalized solution of \(u_{n}\) to get
for every \(h\in W^{2,\infty }(\mathbb {R})\) and for every \(\varphi \in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\cap L^{\infty }(Q)\), \(\varphi _{t}\in L^{p'}(0,T;W^{-1,p'}(\Omega ))\) with \(\varphi (T,x)=0\) such that \(h'(u_{g,n})\varphi \in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\). In addition, using Theorem 2.15, for every \(n\in \mathbb {N}\) and \(k>0\) there exist a sequence of nonnegative measures \((\lambda _{n,k})\in \mathcal {M}_{b}(Q)\) satisfying
- (i)
\(\lambda _{n,k}\underset{\begin{array}{c} n\rightarrow +\infty \\ k\rightarrow +\infty \end{array}}{\longrightarrow }\mu _{s}\) in the narrow topology of measures;
- (ii)
the truncations \(T_{k}(u_{n})\) satisfy
$$\begin{aligned} -\int _{Q}T_{k}(u_{n})v_{t}\mathrm{{d}}x \mathrm{{d}}t+\int _{Q}a_{n}(t,x,\nabla T_{k}(u_{n}))\cdot \nabla v \mathrm{{d}}x\mathrm{{d}}t=\int _{Q}\tilde{v} \mathrm{{d}}\mu _{0}^{n}+\int _{Q}\tilde{v} d\lambda _{n,k}+\int _{\Omega }T_{k}(u_{0}^{n})v(0)\mathrm{{d}}x \end{aligned}$$(3.6)for every \(v\in W\cap L^{\infty }(Q)\) such that \(v(T)=0\) (with \(\tilde{v}\) being the unique \(\text {cap}_{p}\)-quasi continuous representative of v).
Remark 3.2
If we prove that \((u_{n})\) converges to u a.e. in Q and that u is a renormalized solution to (3.2). Then using Theorem 2.15 and for every \(k>0\), the truncation functions are variational solutions of problems
and
for suitable measures \(\mu _{n,k}\), \(\mu _{k}\in \mathcal {M}_{0}(Q)\). Let us finally remark that Eqs. (3.3)–(3.4) are not consequences of the G-convergence of the operators because of the varying right-hand sides \(\mu _{n,k}\) converging in the weak topology to \(\mu _{k}\).
4 Some a priori estimates and convergence results
Let us choose \(h(u_{n})=T_{k}(u_{n})\) with \(\varphi \equiv 1\) in (3.5). Then for every \(n\in \mathbb {N}\), we have
so that using assumption (2.3) and [1, Prop. 5.2], we get
Now, by using [29, Proposition 3.1] and the estimate (4.2), there exists \(C>0\), independent of k and n, such that
where \(\mathcal {L}^{N}\) denotes the N-dimensional Lebesgue measure. Thanks to the second inequality of (4.3)
where \(\overline{C}>0\) depends on q and not on n (see [59, Sect. 2.2]).
As a consequence of the above estimates we obtain the following theorem.
Theorem 4.1
Let \(u_{n}\) be a sequence of renormalized solutions of problem (3.1). Then there exist a measurable function \(u:Q\rightarrow \mathbb {R}\) finite a.e. in Q such that (up to subsequences)
- (i)
\(u_{n}\) converges to u a.e. in Q, \(u\in \mathcal {T}^{1,p}_{0}(Q)\) and \(|\nabla u|^{p-1}\in L^{q}(Q)\) for every \(1\le q<p-\frac{N}{N-1}\);
- (ii)
\(T_{k}(u_{n})\) converges to \(T_{k}(u)\) weakly in \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) and there exists \(C=C(q)>0\) such that
$$\begin{aligned} \int _{Q}|\nabla u|^{q(p-1)}\mathrm{{d}}x \mathrm{{d}}t\le C,\quad \forall q<\frac{Np-N+p}{(N+1)(p-1)}; \end{aligned}$$(4.5) - (iii)
\(T_{k}(u_{n})\) converges to \(T_{k}(u)\) strongly in \(L^{2}(Q,\mu _{0})\) and \(\mu _{0}\)-a.e. in Q;
- (iv)
\(\exists \sigma \in (L^{q}(Q))^{N}\) for every \(q<p-\frac{N}{N+1}\), such that \(\sigma _{k}=\sigma \chi _{\lbrace |u|<k\rbrace }\in (L^{p'}(Q))^{N}\) for a.e. \(k>0\), and \((a_{n}(t,x,\nabla u_{n}))_{n\in \mathbb {N}}\) converges to \(\sigma \) weakly in \((L^{q}(Q))^{N}\) for every \(q<p-\frac{N}{N+1}\), while \((a_{n}(t,x,\nabla T_{k}(u_{n})))_{n\in \mathbb {N}}\) converges to \(\sigma _{n}\) weakly in \((L^{p'}(Q))^{N}\).
Proof
The convergence results (i)–(ii) are obtained through similar arguments of [1, 61] under the same assumptions but for fixed operators. To see that (iii) is true, it is enough to use (ii) and Lemma 2.7. Now, by setting \((u_{n})\) and u be such that (i)–(iii) hold, from (2.4) and (4.4) we have \(a_{n}(t,x,\nabla u_{n})\) is bounded in \((L^{q}(Q))^{N}\) for every \(q<p-\frac{N}{N+1}\). Then (up to subsequences) there exist a function \(\sigma \in (L^{q}(Q))^{N}\) such that \((a_{n}(t,x,\nabla u_{n}))\) converges to \(\sigma \) weakly in \((L^{q}(Q))^{N}\). Note that (2.4) and (4.2) ensure that there exist a subsequence \((a_{n}(t,x,\nabla T_{k}(u_{n})))\) (depending on k) and a function \(\sigma _{k}\) such that \((a_{n}(t,x,\nabla T_{k}(u_{n}))\) converges to \(\sigma _{k}\) weakly in \((L^{p'}(Q))^{N}\). Thus, since \(a_{n}(t,x,0)=0\) for every \(n\in \mathbb {N}\), we have \(a_{n}(t,x,\nabla T_{k}(u_{n}))=a_{n}(t,x,\nabla u_{n})\chi _{\lbrace |u_{n}|<k\rbrace }\), and, by (i), for almost every \(k>0\) the functions \(\chi _{\lbrace |u_{n}|<k\rbrace }\) converges to \(\chi _{\lbrace |u|<k\rbrace }\) a.e. in Q. It is easy to see that \(\sigma _{k}=\sigma \chi _{\lbrace |u|<k\rbrace }\) by Remark 2.5. Finally, the sequence of subsequences \((a_{n}(t,x,\nabla T_{k}(u_{n})))\) converges to \(\sigma _{k}\) weakly in \((L^{p'}(Q))^{N}\) for every \(k>0\). \(\square \)
Remark 4.2
Observe that the function \(\frac{T_{k}(u_{n})}{k}\in L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) and satisfies \(\frac{T_{k}(u_{n})}{k}=1\) a.e. in \(\lbrace |u_{n}|>k\rbrace \), then by using the result of [64, Theorem 1.2] and the estimate (4.2) we get
5 Some a priori estimates for measures
As we have seen, we provide a different, and in some sense more natural approach, to deal with nonlinear parabolic problems with measures using G-convergence theory. Before passing to the proof of our main result, let us state some interesting a priori estimates for the measures \(\lambda _{n,k}\) using the convergence results of Theorem 4.1.
Lemma 5.1
For every \(\varphi \in C^{1}(\overline{Q})\) and for every \(n\in \mathbb {N}\), there exists \(\omega =\omega (n,k)\) satisfying
Proof
Let \(k>\delta >0\), and let \(S_{\delta ,k},h_{\delta ,k}:\mathbb {R}\rightarrow \mathbb {R}\) be two Lipschitz functions defined by (see Fig. 3)
The functions \(S_{\delta ,k}(s)\) and \(h_{\delta ,k}(s)\)
Since \(h_{\delta ,k}(u_{n})\varphi \) is an admissible test function both in (3.5) and (3.6) for every \(\varphi \in C^{1}(\overline{Q})\) with \(\mu _{0}=f-\text {div}(G)+g_{t}\), so that using (3.5) we get
where \(H_{\delta ,k}(s)=\int _{0}^{s}h_{\delta ,k}(r)dr\). On the other hand (3.6) implies
It is easy to prove that the first and last terms are in fact equivalent to \(\omega (n,k)\) if we use the convergence in \(L^{1}(Q)\) of \(u_{n}\), \(|a_{n}(t,x,\nabla u_{n})|\), \(|a_{n}(t,x,\nabla T_{k}(u_{n}))|\) and the properties of \(\varphi \)
Which yields
and hence, for \(q<p-\frac{N}{N+1}\),
for every \(\varphi \in C^{1}(\overline{Q})\). Similarly we get
which is a consequence of (4.3), the absolute continuity of the Lebesgue measure (concerning the term \(\Vert \nabla \varphi \Vert _{(L^{q'}(\lbrace |u_{n}|\ge k))^{N}})\) and Remarks 2.2 and 4.2 (for the term \(|\mu _{0}|(\lbrace |u_{n}|\ge k\rbrace ))\). Thus if we consider test functions \(T_{k}(u_{n})\) in (3.6), we have from (4.1)
Then there exist a sequence of nonnegative measures \(\lambda _{k}\in \mathcal {M}_{b}(Q)\) such that (up to a subsequence) the sequences \((\lambda _{n,k})\) converges to \(\lambda _{k}\) in the weak-* topology of \(\mathcal {M}_{b}(Q)\) as n goes to \(+\infty \). Moreover, a passage to the limit on n in (5.1) gives
for every \(\varphi \in C^{\infty }_{c}(Q)\), that is the sequence \((\lambda _{k})\) converges to \(\mu _{s}\) in the weak-* topology of \(\mathcal {M}_{b}(Q)\) as k goes to \(+\infty \). \(\square \)
The reconstruction property of the sequence \((\lambda _{k})\) is essentially played by a technical Lemma.
Lemma 5.2
Let u and \(\sigma \) be the functions introduced in Theorem 4.1, and let \(\lambda _{k}\in \mathcal {M}_{b}(Q)\) be the measures introduced above. The \(\lambda _{k}\) belongs to \(\mathcal {M}_{0}(Q)\), and
for every \(v\in W\cap L^{\infty }(Q)\) such that \(v(T)=0\) and for a.e. \(k>0\). Moreover there exists a nonnegative measure \(\gamma \in \mathcal {M}_{b}(Q)\) independent of k such that \(\lambda _{k}-\gamma \) belong to \(\mathcal {M}_{0}(Q)\), and
for every \(v\in W\cap L^{\infty }(Q)\) such that \(v(T)=0\) and for a.e. \(k>0\).
Proof
For every \(k>0\), by Theorem 4.1, \((a_{n}(t,x,\nabla T_{k}(u_{n})))\) converges to \(\sigma \chi _{\lbrace \vert u\vert <k\rbrace }\) weakly in \((L^{p'}(Q))^{N}\), \((\chi _{\lbrace |u_{n}|<k\rbrace })\) converges to \(\chi _{\lbrace |u|<k\rbrace }\)\(\mu _{0}\)-a.e. in Q, and by the fact that \(\lambda _{k}\) is the weak-* limit of \(\lambda _{n,k}\), by passing to the limit in (3.6) as \(n\rightarrow +\infty \) for every test function \(\varphi \in C^{\infty }_{c}(Q)\), we get
Using the fact that \(\sigma \chi _{\lbrace |u|<k\rbrace }\) belongs to \((L^{p'}(Q))^{N}\), the measure \(\lambda _{k}\) belongs to \(\mathcal {M}_{0}(Q)\) and (5.12) can be extended to every test function \(\varphi \in W\cap L^{\infty }(Q)\) such that \(\varphi (T)=0\) by using a standard density argument. Now suppose that the Lebesgue measure of the set \(\lbrace u(t,x)=0\rbrace \) is zero (if it’s not, we can replace \(u=0\) with \(u=a\) (a is a nonnegative value) where \(\mathcal {L}^{N}(\lbrace u=a\rbrace )\)), so that for \(\delta >0\) and for the Lipschitz function \(h_{\delta }(s):\mathbb {R}\rightarrow \mathbb {R}\) defined by (see Fig. 4).
The function \(h_{\delta }(s)\)
If we choose \(h_{\delta }(u_{n})\varphi \), with \(\varphi \in C^{\infty }_{c}(Q)\), as test function in (3.6) for \(k>\delta \), we have
Using also (2.3) and (4.1), for every \(\delta >0\) and for every \(n\in \mathbb {N}\)
then there exists a sequence \( \delta _{h} \ \overrightarrow{h \rightarrow \infty } \ 0 \), and a nonnegative measure \(\gamma _{n}\in \mathcal {M}_{b}(Q)\) such that
for every \(\varphi \in C^{\infty }_{c}(Q)\). Moreover \(0\le \gamma _{n}(Q)\le |\mu _{n}|(Q)+|u_{0}^{n}|(\Omega )\), so that, up to subsequences, \(\gamma _{n}\) converges to a nonnegative \(\gamma \) in the weak-* topology of \(\mathcal {M}_{b}(Q)\) allows to pass to the limit in (5.14) as \(\delta \rightarrow 0\) to obtain
Due to the passage to the limit as n tends to \(+\infty \), we conclude
for every \(\varphi \in C^{\infty }_{c}(Q)\). Recall that, since \(\sigma \chi _{\lbrace 0<u<k\rbrace }\) belongs to \((L^{p'}(Q))^{N}\), the measure \(\lambda _{k}-\delta \) belongs to \(\mathcal {M}_{0}(Q)\) and (5.15) holds for every test function in \(W\cap L^{\infty }(Q)\). \(\square \)
6 Proof of Theorem 3.1
Thanks to the above estimates we are able to prove Theorem 3.1. For the sake of simplicity, in what follows, the convergences are all understood to be taken up to a suitable subsequence extraction, even if no explicitly claimed. As usual, \(u_{n}\) and u will be respectively the sequence of renormalized solutions of the associated problems such that all the results in Sects. 4 and 5 hold.
Step .1 The limit equation. Let \(w\in W\cap L^{\infty }(Q)\) be fixed, we take \(v=T_{k}(u_{n})-w\) as test function in (3.6) to obtain
Using Lemma 1.4, we can replace \(T_{k}(u_{n})_{t}\) with \(w_{t}\), we get
Setting \(\varphi \in C^{\infty }_{c}(Q)\) and using \(w_{n}\) as variational solutions to
The hypothesis on G-convergence of the operators implies that \(w_{n}\) converges weakly in \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\) to the unique solution \(w_{0}\) of
that is \(w_{n}\) converges to \(\varphi \) weakly in \(L^{p}(0,T;W^{1,p}_{0}(\Omega ))\). Moreover, since \(a_{0}(t,x,\nabla \varphi )\) belongs to \((L^{\infty }(Q))^{N}\), the regularity results of Remark 2.11 imply that \((w_{n})\) is equi-Hölder continuous, and hence converges uniformly to \(\varphi \) in Q . Now we choose \(w=w_{n}\) in (6.2) in order to get
Using the equation solved by \(w_{n}\) and (6.5), we have
which allows to pass to the limit in (6.6) as n goes to \(+\infty \) to obtain
Recall that in the last two terms we use the lower semi-continuity of the masses of weakly-* converging measures, so that we have from (6.7)
which yields, by Lemma 2.12
for every \(\varphi \in C^{\infty }_{c}(Q)\). By density arguments and since \(\lambda _{k}\in \mathcal {M}_{0}(Q)\), (6.9) is valid for test function in \(W\cap L^{\infty }(Q)\). In particular, for \(\varphi =T_{k}(u)-v\), \(v\in W\cap L^{\infty }(Q)\), we obtain
Then by the characterization of Theorem 2.15, u is a renormalized solution of (3.2) where \(\lambda _{k}\) converge to \(\mu _{s}\) in the narrow topology of measures (see Step. 3). So that, by choosing \(v=h(u)\varphi \), \(u\in W^{1,\infty }(\mathbb {R})\) and \(\varphi \in C^{\infty }_{c}(Q)\) in (6.10), an easy passage to the limit as \(k\rightarrow \infty \) leads to
Step 2. Convergence of the momenta. Hereafter, we study the limit of the sequence \(a_{n}(t,x,\nabla v_{n})\), this is done by having \(v_{n}\) as variational solution of
where \(\eta \) is a fixed element of \(\mathbb {R}^{N}\), we take advantage of G-convergence properties to get
In addition, by Remark 2.11, \((v_{n})\) is equi-Hölder continuous, we can assume that
The monotonicity assumption (2.5) implies
for every \(\varphi \in C^{\infty }_{c}(Q)\) with \(\varphi \ge 0\). In order to pass to the limit in (6.13), we use the limit integral
by compensated compactness (see [51, 77]). To complete the passage to the limit in (6.13) we establish the same result for the term
where the sequence \(\text {div}\left( a_{n}(t,x,\nabla T_{k}(u_{n}))\right) \) converges in a weak sense, and we get
using the formulation (3.6), we obtain
Therefore, as \(n\rightarrow +\infty \) and using the dominated convergence Theorem for the first integral, the fact that \(v_{n}\) converges uniformly in Q and that the measures \(\lambda _{k,\eta }\) converge weak-* in \(\mathcal {M}_{b}(Q)\) in other integrals, we obtain
where \(\sigma _{k}\) is given in Theorem 4.1 (iv).
Now, since \(T_{k}(u_{n})-v_{n}\) converges strongly to \(T_{k}(u)-(t,x)\cdot \eta \) in \(L^{p}(Q)\), \(a_{n}(t,x,\nabla T_{k}(u_{n}))\) converges weakly in \((L^{p'}(Q))^{N}\) to \(\sigma _{k}\) and using (6.15) and (6.17), we have
that, together with (6.14) and using also the limit equation of (6.13):
for every \(\varphi \in C^{\infty }_{c}(Q)\) with \(\varphi \ge 0\). Hence, for every \(\eta \in \mathbb {R}^{N}\) there exists a set \(E(\eta )\subseteq Q\) with Lebesgue measure zero such that
Now, consider \(E=\cup _{m}E(\eta _{m})\) where \((\eta _{m})\) is a countable dense set of \(\mathbb {R}^{N}\), we have
in view of the continuity of \(a_{0}(t,x,\cdot )\), we obtain
Note that \(a_{0}(t,x,\cdot )\) is continuous and monotone in \(\mathbb {R}^{N}\) with the fact that (6.19), we have \(\sigma _{k}(t,x)=a_{0}(t,x,\nabla T_{k}(u))\) a.e. in Q. Using Theorem 4.1 (iv), we deduce that \((a_{n}(t,x,\nabla T_{k}(u_{n})))\) converges to \(a_{0}(t,x,\nabla T_{k}(u))\) weakly in \((L^{p'}(Q))^{N}\) and this concludes the proof of Step 2.
Step 3. End of the proof. Let us prove that \(\lambda _{k}\) converges to \(\mu _{s}\) in the narrow topology of measures. Using estimate (5.9), it is easy to prove that \(\lambda _{k}\) converge to \(\mu _{s}\) in the weak-* topology of measures. As a consequence of Remark 2.9, the narrow convergence follows from the convergence of the measures. Then it’s enough to check, since we have \(\mu _{s}(Q)\le \underset{k\rightarrow +\infty }{\text {lim inf }}\lambda _{k}(Q)\) because of the weak-* convergence,
Let us define the Lipschitz function \(h_{k}(s):\mathbb {R}\rightarrow \mathbb {R}\), \(k>0\), by (see Fig. 5).
The function \(h_{k}(s)\)
This step consists in taking \(h_{k}(u)\) as test function in (6.10) corresponding to 2k
Since \(h_{k}(u)\) tend to zero \(\mu _{0}\)-a.e. in Q and by dominated convergence Theorem, this leads to
Due to the definition of \(h_{k}\) (i.e. \(h_{k}(0)=0\)), we can take \(h_{k}(u_{n})\) as test function in (3.5), we have
letting \(n\rightarrow \infty \) then yields
Now we prove that
for every \(\varphi \in C^{\infty }_{c}(Q)\). It’s enough to take \(h_{k}(u_{n})\varphi \) as test function in (3.5) and \(h_{k}(u)\varphi \) as test function in (6.11). Subtracting the two equations we obtain
Comparing (6.10) with (5.10) we deduce that \(a_{0}(t,x,\nabla T_{k}(u))=\sigma \chi _{\lbrace |u|<k\rbrace }\) for a.e. \(k>0\); then, by Theorem 4.1 (iv), the sequence \((a_{n}(t,x,\nabla u_{n}))\) converges to \(a_{0}(t,x,\nabla u)\) weakly in \((L^{q}(Q))^{N}\) for \(q<p-\frac{N}{N+1}\). By definition of \(h_{k}(s)\), \(h_{k}(u_{n})\) converges to \(h_{k}(u)\) both a.e. and \(\mu _{0}\)-a.e. in Q, we can pass to the limit as \(n\rightarrow \infty \) to get (6.26), which implies the weak-* convergence of the sequence of non-negative measures \((\frac{1}{k}a_{n}(t,x,\nabla u_{n}))\cdot \nabla u_{n}\chi _{\lbrace k<u_{n}<2k\rbrace })\) to the nonnegative measure \(\frac{1}{k} a_{0}(t,x,\nabla u)\cdot \nabla u\chi _{\lbrace k<u<2k\rbrace }\). By means of the semi-continuity of the masses, we get
which yields, from (6.23) and (6.25)
this proves the assertion (i) of the narrow convergence, and the fact that u is a renormalized solution then follows straightforwardly, so that the proof of Theorem 3.1 is complete.
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Abdellaoui, M., Azroul, E. Homogenization of a nonlinear parabolic problem corresponding to a Leray–Lions monotone operator with right-hand side measure. SeMA 77, 1–26 (2020). https://doi.org/10.1007/s40324-019-00197-8
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DOI: https://doi.org/10.1007/s40324-019-00197-8