Weak Structural Stability of Pseudo-Monotone Equations

DISCRETE AND CONTINUOUS doi:10.3934/dcds.2015.35.2763 DYNAMICAL SYSTEMS Volume 35, Number 6, June 2015 pp. 2763–2796

WEAK STRUCTURAL STABILITY OF PSEUDO-MONOTONE EQUATIONS

Augusto Visintin Universit`adegli Studi di Trento Dipartimento di Matematica via Sommarive 14 38050 Povo (Trento), Italy

Dedicated to J¨urgen Sprekels on the occasion of his 65th birthday

Abstract. The inclusion β(u) 3 h in V 0 is studied, assuming that V is a reﬂexive Banach space, and that β : V → P(V 0) is a generalized pseudo- monotone operator in the sense of Browder-Hess [MR 0365242]. A notion of strict generalized pseudo-monotonicity is also introduced. The above inclusion is here reformulated as a minimization problem for a (nonconvex) functional V × V 0 → R ∪ {+∞}. A nonlinear topology of weak-type is introduced, and related compactness results are proved via De Giorgi’s notion of Γ-convergence. The compactness and the convergence of the family of operators β provide the (weak) structural stability of the inclusion β(u) 3 h with respect to variations of β and h, under the only assumptions that the βs are equi-coercive and the hs are equi-bounded. These results are then applied to the weak stability of the Cauchy problem for doubly-nonlinear parabolic inclusions of the form Dt∂ϕ(u) + α(u) 3 h, ∂ϕ being the subdiﬀerential of a convex lower semicontinuous mapping ϕ, and α a generalized pseudo-monotone operator. The technique of compactness by strict convexity is also used in the limit procedure.

1. Introduction. Several models may be reduced to doubly-nonlinear parabolic equations of the form

Dt∂ϕ(u) + α(u) 3 h (Dt := ∂/∂t); (1.1) here ∂ϕ is the subdifferential of a convex and lower semicontinuous mapping ϕ, and α is a (nonlinear elliptic) differential operator with some compactness and monotonicity properties. For instance, in several models of diffusion and in electro- magnetism one respectively encounters operators of the form α(u) = −∇·f(u, ∇u)(∇·:= div), (1.2) α(u) = ∇×f(u, ∇×u)(∇×:= curl). (1.3)

2010 Mathematics Subject Classiﬁcation. 35K60, 49J40, 58E. Key words and phrases. Pseudo-monotone operators, Γ-convergence, structural stability, variational formulation, quasilinear parabolic equations. The author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilit`ae le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This work was partially supported by the MIUR-PRIN 10-11 grant for the project “Calculus of Variations”. The author gratefully acknowledges several helpful remarks of the anonymous Reviewers.

2763 2764 AUGUSTO VISINTIN

If f is continuous in the first argument and monotone in the second one, then these two operators are pseudo-monotone in the sense of Brezis [9]. Weak Structural Stability. In this work we are concerned with the dependence of the solution u on the mapping ϕ, on the operator α (i.e., on the function f in the case of (1.2) or (1.3)), and on the source term h. More generally, let V be a reflexive Banach space, and let us consider a problem of the form β(u) 3 h (with β : V → P(V0) and h ∈ V0). (1.4) For instance, β may represent a nonlinear differential operator that acts on functions that depend either on x or on (x, t). In this work we address these two issues: (i) Compactness of the operators: we devise a (nonlinear) notion of convergence for operators, that grants that each equi-coercive sequence {βn} has a cluster point. Provisionally, let us label this convergence by τ. (ii) Convergence of the solutions: we prove that, if the operators βn τ-converge 0 and the data hn converge in V , then the corresponding solutions un weakly converge to a solution of the asymptotic problem, up to extracting a subsequence. 1 Dealing with problems with low regularity, we must cope with weak topologies. We prove that 2 0 if βn(un) 3 hn ∀n and hn → h in V , then, extracting subsequences that we label by n0, 0 (i) ∃β : V → P(V ) such that βn0 →τ β, (1.5)

(ii) ∃u ∈ V such that un0 * u in V, (iii) β(u) 3 h. One may interpret this conclusion as the (weak) structural stability of the problem (1.4). The selection of an appropriate topology τ is the key issue in this program, since compactness of the operators and convergence of the solutions are compet- ing requirements. The identiﬁcation of the properties of the limit operators β is especially relevant. 0 For any set A ⊂ V×V we set IA(v) = 0 if v ∈ A and IA(v) = +∞ otherwise. We shall denote the graph of any operator β : V → P(V0) by G(β), and the indicator function of this graph by IG(β). At the basis of our analysis there is the reformulation of the inclusion β(u) 3 h as the null-minimization problem 3 ∗ ∗ 0 IG(β)(u, h) = inf IG(β)(v, v ):(v, v ) ∈ V×V = 0, (1.6) and then the use of De Giorgi’s theory of Γ-convergence w.r.t. a nonlinear topology of weak-type. This simple expedient is equivalent to the Kuratowski convergence of the graphs w.r.t. that topology. Doubly-Nonlinear Cauchy Problems. After developing the theory in general terms, we shall apply our results to the structural stability of a family of doubly- nonlinear Cauchy problems of the form 0 p 0 Dt∂ϕ(u) + α(u) 3 h in V = L (0,T ; V ) , 0 (1.7) ∂ϕ(u) t=0 3 w ;

1 Throughout this paper the term asymptotic will always refer to the limit behavior along a sequence, rather than to diverging time. 2 We denote the strong, weak, and weak star convergence respectively by →, *, *∗ . 3 The use of indicator functions allows one to reduce any equation to variational form: β(u) = h is tantamount to IG(β)(u, h) = inf IG(β)(·, h). (This author hopes that this not-very-honorable trick has already been used in the past, so that it may be ascribed to someone else.) PSEUDO-MONOTONE EQUATIONS 2765 here ϕ is a convex and lower semicontinuous mapping, 1 < p < +∞, and V is a 1,p reflexive Banach space (e.g., V = W0 (Ω)). In several cases the operator α : V → P(V0) is maximal monotone in the sense of Minty [36] and Browder [12]. In other cases α is single-valued and pseudo-monotone in the sense of Brezis [9]. However, as a sequence of single-valued operators may easily converge to a multi-valued one, just families of multi-valued pseudo-monotone operators may be expected to be compact. This induces us to deal with the more general class of the generalized pseudo- monotone operators in the sense of Browder and Hess [13], which may actually be multi-valued. Structure of the Present Work. The following procedure is at the basis of this work: (i) the equation is reformulated as a minimization problem; (ii) loosely speaking, a suitable asymptotic property of weak semicontinuity is encoded into the topology of the function space. More specifically, this property is the sequential weak upper semicontinuity of the duality product, that occurs as a hypothesis in the definition of generalized asymptotic pseudo-monotonicity; (iii) the Γ-compactness of the minimized functionals is proved with respect to that topology; (iv) the G-compactness of the equation is then derived. One may distinguish three parts in this paper. In Sect. 2–4 we illustrate some notions related to pseudo-monotonicity; in Sect. 5 and 6 we deal with compactness of certain families of operators; in Sect. 7 and 8 we apply the above results to equations of the form (1.1). More specifically, in Sect. 2 we review the notions of pseudo-monotonicity and generalized pseudo-monotonicity. This property combines monotonicity and compactness properties for multi-valued operators, and encompasses a wide class of quasilinear PDEs. We then introduce the notion of strict generalized pseudo- monotonicity, which has the advantage of providing strong convergence, see Theo- rem 2.2. In Sect. 3 we deal with families of operators and introduce two properties, that we name asymptotic lower-semicontinuity, and uniformly strict asymptotic lower- semicontinuity. They might be regarded as sequential analogs of the notions of generalized pseudo-monotonicity and strict generalized pseudo-monotonicity. In Theorem 3.3 we derive a result of convergence in norm via compactness by strict convexity, in the sense of [45, 46]. In Sect. 4 we exhibit some examples of asymptotically lower-semicontinuous families of operators in the class of quasilinear elliptic operators. As we saw, the analysis of structural stability raises the necessity of proving the compactness of families of operators. To this purpose, first we need a suitable notion of convergence. In Sect. 5 we define two nonlinear weak-type topologies on 0 V×V , that we label by πe and πe+ (see (5.2), (5.3)). In Theorem 5.5 we show that any equi-coercive family of functionals ψ : V×V0 → R∪{+∞} is sequentially Γ-compact with respect to either of the topologies πe and πe+. This result rests upon the theory of Γ-convergence; see also [2]. Here account is taken of the possibility that bounded sets in the topology πe+ fail to be metrizable; this requires a careful extension of Γ-compactness results of [47], that rests on a technique of [2, 17]. 2766 AUGUSTO VISINTIN

In Sect. 6 we draw consequences for sequences of operators, by identifying each operator α with its graph G(α) and with the corresponding indicator function IG(α):

0 0 0 α : V → P(V ) ⇔ G(α) ⊂ V×V ⇔ IG(α) : V×V → R ∪ {+∞}. (1.8)

In this way we are able to express the convergence of the operators via the Γ- convergence of the indicator functions of the associated graphs. In Theorems 6.2 and 6.3 we prove that, if a family of operators α : V → P(V0) is equi-coercive and asymptotically lower-semicontinuous, then the sequence of the graphs is sequentially compact in the sense of Kuratowski w.r.t. the topology πe+, and any cluster operator is generalized pseudo-monotone, see Corollary 6.4. Next we apply the above results to the structural stability of a family of (semi-)flows of the form (1.7). In Sect. 7 we prove existence of a solution assuming that α is generalized pseudo-monotone. This is preparatory for the next section, which deals with compactness and structural stability of the doubly-nonlinear inclusion β(u) := Dt∂ϕ(u) + α(u) 3 h, in the sense of (1.5). In order to overcome the lack of monotonicity, among other things here we use a nonstandard estimate procedure, which may be found e.g. in [1, 46]. In Sect. 8 we deal with the structural stability of the Cauchy problem for the system (1.7): we let the data vary along three sequences {hn}, {ϕn} and {αn}, and derive an asymptotic problem. This result is not straightforward, since by our stipulation we may just assume the convergence properties that stem from the uniform coerciveness of the operators and from the boundedness of the solution in natural function spaces. The weakness of these convergences makes this limit analysis nontrivial. More specifically, first in Theorem 8.3 we derive a weaker asymptotic formulation than (1.7), with a limit operator that a priori might exhibit memory. This raises the question of showing that a subsequence of the solutions {un} and a suitable 0 selection {zn ∈ αn(un)} weakly converge respectively in V and V for a.e. t. In Theorem 8.4 we assume that the αns are of the form (1.2) (or (1.3)), and overcome this difficulty via a technique of compactness by strict convexity. We emphasize that this does not require the operators αn’s and α to be either strongly monotone or Lipschitz-continuous. In the case of operators of the form (1.2) or (1.3), our hypotheses are thus consistent with the occurrence of either horizontal or vertical inflection points in the graph of the functions ξ 7→ f(x, v, ξ), for a.e. x and any v. This thus extends previously known results, see e.g. [37] and references therein. Comparison with Other Works. The notion of pseudo-monotonicity is due to Brezis [9], see also e.g. [12, 33, 50], and was extended to that of generalized pseudo-monotonicity by Browder and Hess [13]. Our compactness and convergence results may be labelled as G-convergence. This notion was introduced by Spagnolo in [41] for linear elliptic and parabolic operators; in [15] G-convergence and G-compactness were then extended to quasilinear elliptic problems in divergence form. The related notion of H-convergence was introduced by Tartar in the homogenization of linear elliptic equations [42]; see the monograph [43] for a detailed account. This theory was applied to a number of homogenization problems, and was extended to quasilinear elliptic problems in divergence form in [22]. The problem of homogenization (see e.g. [5, 16, 29]) has indeed several elements of similarity with structural stability. PSEUDO-MONOTONE EQUATIONS 2767

The present treatment rests upon a different approach, and differs from the above-mentioned works on G- and H-convergence in the following aspects: (i) a class of (either stationary or evolutionary) abstract equations is considered; (ii) the problem is reformulated as a minimization principle; (iii) Γ-compactness and Γ-convergence with respect to a nonlinear topology are used; (iv) these results are applied to quasilinear parabolic problems, in which the elliptic part is generalized pseudo-monotone. The homogenization of quasilinear elliptic equations was studied in several works, see e.g. [6, 23, 24, 35, 38]. Quasilinear parabolic equations were also treated in the monograph [37], see also references therein. De Giorgi’s theory of Γ-convergence (see e.g. the pioneering work [18] and the monographs [3,7,8, 17]) was used for cyclically maximal monotones operators, starting with [35]. The structural stability of doubly-nonlinear parabolic equations was also studied in [39], without addressing the compactness of the family of operators. A more specific comparison with the recent articles [47, 48, 49] of this author seems also in order. In [47] the representation due to Fitzpatrick [21] was used for the variational formulation of maximal monotone flows, and was then applied to the structural stability of quasilinear PDEs with a single nonlinearity. In [48] this method was used in the homogenization of a quasilinear model of Ohmic electric conduction with Hall effect. In the present work we do not use the Fitzpatrick theory. Dealing with pseudo- monotone operators, as a potential here instead we use the indicator function of the operator, which fails to be convex for nonlinear operators. This nonconvexity is a main source of difficulty for the present work. Concerning applications to PDEs, we however improve the results of [47], by encompassing a larger family of equations and by getting stronger existence theorems for the Cauchy problem. A different approach to pseudo-monotone operators, based on an extension of the Fitzpatrick theory, is worked out in [49].

2. Strict generalized pseudo-monotonicity. In this section first we review the notions of pseudo-monotonicity and of generalized pseudo-monotonicity. We then introduce what we name strict generalized pseudo-monotonicity, and exhibit some examples. Pseudo-Monotone Operators. Let V be a reflexive and separable real Banach space; let us denote by k·k its norm and by h·, ·i the duality pairing between V 0 and V . Along the lines of [9] and [33, p. 180], we shall say that an everywhere-defined single-valued operator α : V → V 0 is pseudo-monotone if

(i) α is bounded (2.1)

(i.e., it maps bounded subsets of V to bounded subsets of V 0), and

(ii) ∀u ∈ V, ∀ sequence {un} in V,

un * u, lim supn→∞ hα(un), un − ui ≤ 0 (2.2)

⇒ hα(u), u − vi ≤ lim infn→∞ hα(un), un − vi ∀v ∈ V, 2768 AUGUSTO VISINTIN or equivalently, see e.g. [33],

(ii)’ ∀u ∈ V, ∀ sequence {un} in V,

un * u, lim supn→∞ hα(un), un − ui ≤ 0 (2.3)

⇒ α(un) * α(u), hα(un), un − ui → 0.

Examples of Pseudo-Monotone Operators. Here we brieﬂy review some classical examples. (i) Hemicontinuous monotone operators, compact operators, as well as sums of operators of these classes, are pseudo-monotone; see e.g. [9, 33]. (ii) Whenever A : V 2 → V 0 is such that

v 7→ A(u, v) is bounded, monotone and hemicontinuous, ∀u ∈ V, (2.4) u 7→ A(u, v) is continuous from V -weak to V 0-strong, ∀v ∈ V, the operator α : V → V 0 : u 7→ A(u, u) is pseudo-monotone, see e.g. [9, 12, 33, 50]. (iii) A wider class of pseudo-monotone operators is provided by the operators of the calculus of variations; see e.g. [33, Sect. 2.2.5]. This class was introduced by Leray and Lions [32], and was applied to the analysis of quasilinear elliptic PDEs e.g. in [25]. (iv) This and the next example display two relevant classes of diﬀerential operators of the calculus of variations. Let Ω be a bounded Lipschitz domain of RN , and f :Ω×R×RN → RN be such that

∀(v, ξ), the function x 7→ f(x, v, ξ) is Lebesgue-measurable, (2.5) for a.e. x, ∀ξ, the function v 7→ f(x, v, ξ) is continuous, (2.6) for a.e. x, ∀v, the function ξ 7→ f(x, v, ξ) is monotone, (2.7)

∃c1, c2 > 0 : for a.e. x, ∀(v, ξ), |f(x, v, ξ)| ≤ c1(|v| + |ξ|) + c2. (2.8)

The operator

0 1 αf (u) = −∇ · [f(x, u, ∇u)] in D (Ω), ∀u ∈ H0 (Ω) (2.9) is then an operator of the calculus of variations. (v) Let us assume that N = 3, p = 2, and that f is as above. As the operator v 7→ ∇×[f(x, v, ∇×v)] is typically restricted to divergence-free functions, we deﬁne the Hilbert spaces

H := v ∈ L2(Ω)3 : ∇·v = 0 in D0(Ω), ν·v = 0 in H−1/2(∂Ω) , (2.10) V := H ∩ H1(Ω)3. Here ν is the outward-oriented unit vector-ﬁeld on ∂Ω, and v 7→ ν·v is meant in the sense of the traces, see e.g. [34]. It is known that H ⊂ ∇×H1(Ω)3, under suitable restrictions on the domain Ω; this holds e.g. if Ω is diﬀeomorphic to a convex set, see e.g. [44]. By identifying H with its dual H0, we then get the Hilbert triplet

V ⊂ H = H0 ⊂ V 0 with continuous and dense injections. (2.11)

It is not diﬃcult to check that

0 3 αef (u) = ∇×[f(x, u, ∇×u)] in D (Ω) , ∀u ∈ V (2.12) PSEUDO-MONOTONE EQUATIONS 2769 is also an operator of the calculus of variation. Generalized Pseudo-Monotone Operators. The class of pseudo-monotone operators extends that of single-valued maximal monotone operators; maximal monotone operators however may also be multi-valued. This raises the question of defining multi-valued pseudo-monotone operators. Following [13] (see also e.g. [27, Chap. 3]), we shall say that a multi-valued operator α : V → P(V 0) is generalized pseudo-monotone if, setting G(α) := {(u, u∗): u∗ ∈ α(u)} (the graph of α), ∗ 0 ∗ ∀(u, u ) ∈ V ×V , ∀ sequence {(un, un)} in G(α), ∗ ∗ ∗ un * u, un * u , lim sup hun, un − ui ≤ 0 (2.13) n→∞ ∗ ∗ ⇒ u ∈ α(u), hun, un − ui → 0. The analogy with (2.3) is evident; however here α is not assumed to be bounded. ∗ ∗ As the preceding convergence also reads hun, uni → hu , ui, α is generalized pseudo- monotone if and only if so is α−1. In passing notice that (2.13) entails ∗ ∀u ∈ V, ∀ sequence {(un, un)} in G(α), ∗ ∗ ∗ (2.14) un * u, un * u ⇒ lim supn→∞ hun, un − ui ≥ 0. This class obviously includes (single-valued) pseudo-monotone operators, and also multi-valued maximal monotone operators, see e.g. [10, p. 27]. Other examples of generalized pseudo-monotone operators may be found e.g. in [13, 26, 27, 30, 31, 37]. The surjectivity of generalized pseudo-monotone operators is a nontrivial issue. We just state a result of Browder and Hess, that we shall apply ahead (see also [27, p. 368]). Lemma 2.1. [13] Let an operator α : V → P(V 0) be generalized pseudo-monotone, and assume that α is bounded, (2.15) ∀v ∈ V, α(v) is nonempty, closed and convex, (2.16) ∃k > 0 : ∀(v, v∗) ∈ G(α), hv∗, vi + kkvk ≥ 0, (2.17) ∃c : R+ → R, with c(r) → +∞ as r → +∞, such that (2.18) ∀(v, v∗) ∈ G(α), hv∗, vi ≥ c(kvk)kvk. Then α(V ) = V 0. Strict Generalized Pseudo-Monotonicity. Next we strengthen the notion of pseudo-monotonicity. We suggest to call an operator α : V → P(V 0) strictly generalized pseudo-monotone whenever there exists a mapping γ such that γ : V → R is convex and Gâteaux-differentiable, (2.19) γ0 : V → V 0 is bounded, (2.20)

∀ sequence {vn} in V, vn * v, γ(vn) → γ(v) ⇒ vn → v, (2.21) α − γ0 is generalized pseudo-monotone. (2.22) (Here by γ0 we denote the Gˆateauxderivative of γ; see e.g. [20, Sect. I.5].) The class of strictly generalized pseudo-monotone obviously consists of operators α of the form α = β + γ0, with α generalized pseudo-monotone and γ fulﬁlling (2.19)–(2.21). 2770 AUGUSTO VISINTIN

The condition (2.21) is essentially a hypothesis of strict convexity of γ. For instance, if g : RN → R is a strictly convex Gâteaux-differentiable function such that N p p ∃p ∈]1, +∞], ∃C1,C2 > 0 : ∀ξ ∈ R ,C1|ξ| ≤ g(ξ) ≤ C2|ξ| , then (see [46, p. 253]) (2.19)–(2.21) hold for Z 1,p γ(v) = g(∇v) dx ∀v ∈ V = W0 (Ω). (2.23) Ω For instance this includes the case of γ0(v) = −∇·(|∇v|p−2∇v) in V 0 for any v ∈ V . The next statement may be regarded as a result of compactness by strict convexity, in the sense of [45] and Chap. X of [46]. Theorem 2.2. If an operator α : V → P(V 0) is strictly generalized pseudo- monotone (i.e., there exists a mapping γ that fulfills (2.19)–(2.22)), then ∗ ∀ sequence {(un, un) ∈ G(α)}, ∗ ∗ ∗ ∗ un * u, un * u , lim supn→∞ hun, uni ≤ hu , ui (2.24) ∗ ⇒ u ∈ α(u), un → u. (In particular, α is thus generalized pseudo-monotone.) ∗ Proof. Let (2.19)–(2.22) hold, let {(un, un)} be a sequence in G(α), un * u in V , ∗ ∗ 0 ∗ 0 un * u , γ (un) * ξ in V , and ∗ lim sup hun, un − ui ≤ 0. (2.25) n→∞ By this inequality and by the lower semicontinuity of γ, ∗ 0 lim supn→∞ hun − γ (un), un − ui 0 ≤ − lim infn→∞ hγ (un), un − ui

≤ − lim infn→∞ [γ(un) − γ(u)] ≤ 0. By (2.22) then u∗ − ξ∗ ∈ α(u) − γ0(u), (2.26) ∗ 0 hun − γ (un), un − ui → 0. (2.27) Moreover, by the convexity of γ, (2.25) and (2.27) 0 lim sup γ(un) − γ(u) ≤ lim sup hγ (un), un − ui n→∞ n→∞ 0 ∗ ∗ = lim hγ (un) − un, un − ui + lim sup hun, un − ui ≤ 0. n→∞ n→∞

By the lower semicontinuity of γ, we then infer that γ(un) → γ(u), and by (2.21) 0 ∗ 0 we conclude that un → u. By the continuity of γ then ξ = γ (u). By (2.26) we conclude that u∗ ∈ α(u).

3. Asymptotic properties. In this section we deal with families of operators: we introduce what we name asymptotic lower-semicontinuity, and uniformly strict asymptotic lower-semicontinuity. These might be regarded as sequential analogs of generalized pseudo-monotonicity and strict generalized pseudo-monotonicity. In Theorem 3.3 we derive a result of convergence in norm via compactness by strict convexity, in the sense of [45, 46]. Asymptotic π-Lower-Semicontinuity (“a.l.s.c.”). We still assume that V is a reflexive and separable Banach space. We shall denote by π the duality pairing, and say that a family A of operators V → P(V 0) fulfills the property of asymptotic PSEUDO-MONOTONE EQUATIONS 2771 lower-semicontinuity of the duality pairing, or more simply that it is asymptotically π-lower-semicontinuous (“a.l.s.c.”, for short), if ∗ ∀ sequence {αn} in A, ∀ sequence {(un, un) ∈ G(αn)}, ∗ ∗ ∗ ∗ (3.1) un * u, un * u ⇒ lim infn→∞ hun, uni ≥ hu , ui, or equivalently, ∗ ∀ sequence {αn} in A, ∀ bounded sequence {(un, un) ∈ G(αn)}, ∗ (3.2) un * u ⇒ lim infn→∞ hun, un − ui ≥ 0. This obviously entails that ∗ ∀ sequence {αn} in A, ∀ sequence {(un, un) ∈ G(αn)}, ∗ ∗ ∗ ∗ un * u, un * u , lim sup hun, uni ≤ hu , ui (3.3) n→∞ ∗ ∗ ⇒ lim hun, uni = hu , ui. n→∞ This has an obvious analogy with the notion of generalized pseudo-monotonicity, cf. (2.13). Notice that the operators αns are not assumed to be either pseudo-monotone or bounded; moreover, by the symmetry of the definition (3.1), A is a.l.s.c. if and only if so is the family of the inverse operators {α−1 : α ∈ A}. In particular this applies to sequences of operators. In this case a subsequence 0 must be extracted: a sequence {αn} of operators V → P(V ) is a.l.s.c. if and only if ∗ ∀ subsequence {αn0 }, ∀ subsequence {(un0 , u 0 ) ∈ G(αn0 )}, n (3.4) ∗ ∗ ∗ ∗ un0 * u, un0 * u ⇒ lim infn0→∞hun0 , un0 i ≥ hu , ui. Obviously, a weaker property would be obtained if this implication were assumed for the whole sequence {αn}. Reference to subsequences is of course due to the ubiquitous occurrence of extracted subsequences in the analysis of PDEs. Next we exhibit some sufficient conditions for asymptotic pseudo-monotonicity. Proposition 3.1. Let A and B be two a.l.s.c. families of operators V → P(V 0), and A be bounded, in the sense that S{α(v): v ∈ X, α ∈ A} is bounded, for any bounded set X ⊂ V . (3.5)

The family A + B is then also a.l.s.c. Proof. Let us ﬁx two sequences {αn} ⊂ A ∗ and {βn} ⊂ B, and select any sequence {(un, un) ∈ G(αn +βn)} such that un * u in ∗ ∗ 0 V and un * u in V . By the boundedness of A, there exist two bounded sequences ∗ ∗ 0 {wn} and {zn} in V , such that ∗ ∗ ∗ ∗ ∗ (un, wn) ∈ G(αn), (un, zn) ∈ G(βn), un = wn + zn ∀n. As the families A and B are a.l.s.c., it then follows that ∗ ∗ ∗ lim infn→∞ hun, un − ui = lim infn→∞ hwn + zn, un − ui ∗ ∗ (3.6) ≥ lim infn→∞ hwn, un − ui + lim infn→∞ hzn, un − ui ≥ 0.

Proposition 3.2. Let G be a family of operators V 2 → P(V 0) such that ∀A ∈ G, ∀u, v ∈ V,A(u, v) 6= ∅, (3.7) ∀A ∈ G, ∀u ∈ V, v 7→ A(u, v) is monotone, (3.8)  ∗ 0 ∀ sequence {An} in G, ∀ sequence {(un, un)} in V ×V ,  ∗ ∗ ∗ un * u, un ∈ An(un, u) ∀n, un * u (3.9) ∗ ∗ ⇒ lim infn→∞ hun, uni ≥ hu , ui. 2772 AUGUSTO VISINTIN

For any A ∈ G and any u ∈ V , let us set αA(u) := A(u, u). The family of operators A = {αA : A ∈ G} is then a.l.s.c. ∗ 0 Proof. Let {αAn } be a sequence in A, u ∈ V , and {(un, un)} be a sequence in V ×V such that ∗ un ∈ An(un, un) ∀n, un * u in V.

By (3.7) An(un, u) 6= ∅ for any n, and by (3.8) ∗ ∗ ∗ hun − wn, un − ui ≥ 0 ∀wn ∈ An(un, u), ∀n. (3.10) Therefore (3.9) ∗ ∗ lim inf hun, un − ui ≥ lim inf hwn, un − ui ≥ 0. (3.11) n→∞ n→∞

For instance, the condition (3.9) is obviously fulﬁlled whenever ∀u ∈ V, ∀M > 0, S 0 (3.12) {A(w, u): A ∈ G, kwkV ≤ M} is relatively compact in V . Uniformly Strict Asymptotic π-Lower-Semicontinuity. Next we introduce a further asymptotic notion that we shall apply ahead. We shall say that a family A of operators V → P(V 0) is uniformly strictly asymptotically π-lower-semicontinuous (or “uniformly strictly a.l.s.c.”, for short) if there exists a mapping γ that fulﬁlls (2.19)–(2.21) and such that

the family Ae := {α − γ0 : α ∈ A} is a.l.s.c. (cf. (3.1)). (3.13) The next statement may be regarded as a result of compactness by strict convexity, in the sense of [45] and Chap. X of [46], and may be compared with Theorem 2.2. Theorem 3.3. If A is a uniformly strictly a.l.s.c. family of operators V → P(V 0), then A is a.l.s.c., and ∗ ∀ sequence {αn} of A, ∀ sequence {(un, un) ∈ G(αn)}, ∗ ∗ ∗ ∗ (3.14) un * u, un * u , lim supn→∞ hun, uni ≤ hu , ui ⇒ un → u. Proof. (i) First we show that A is a.l.s.c.. Let (2.19)–(2.21) and (3.13) be fulﬁlled. ∗ For any sequence {αn} of A and any bounded sequence {(un, un) ∈ G(αn)} we have ∗ lim infn→∞ hun, un − ui ∗ 0 0 (3.15) ≥ lim infn→∞ hun − γ (un), un − ui + lim infn→∞ hγ (un), un − ui.

If un * u then by (3.13) the ﬁrst addendum at the right side is nonnegative; on the other hand 0 lim inf hγ (un), un − ui ≥ lim inf [γ(un) − γ(u)] ≥ 0. n→∞ n→∞ ∗ By (3.15) then lim infn→∞ hun, un − ui ≥ 0. Thus A is a.l.s.c.. (ii) Next we prove (3.14). Let (2.19)–(2.21) and (3.13) be fulﬁlled. For any ∗ sequence {αn} of A, any u ∈ V and any bounded sequence {(un, un) ∈ G(αn)}, 0 lim sup γ(un) − γ(u) ≤ lim sup hγ (un), un − ui n→∞ n→∞ 0 ∗ ∗ ≤ lim sup hγ (un) − un, un − ui + lim sup hun, un − ui (3.16) n→∞ n→∞ ∗ 0 ∗ = − lim inf hun − γ (un), un − ui + lim sup hun, un − ui. n→∞ n→∞ PSEUDO-MONOTONE EQUATIONS 2773

0 If un * u then, as A − γ is a.l.s.c., it follows that ∗ 0 lim inf hun − γ (un), un − ui ≥ 0. n→∞ ∗ If we also assume that lim supn→∞ hun, un − ui ≤ 0, then by (3.16) we get

lim sup γ(un) ≤ γ(u). n→∞

By the lower semicontinuity of γ(u), then γ(un) → γ(u). By (2.21) we conclude that un → u. (3.14) is thus established.

4. Some diﬀerential examples. In this section we exhibit some examples of families of quasilinear elliptic second-order operators that are asymptotically π- lower-semicontinuous (or “a.l.s.c.”, for short) in the sense of (3.1). For any Euclidean domain D, we shall denote by L(D)(B(D), resp.) the σ- algebra of the Lebesgue- (Borel-, resp.) measurable subsets of D. We shall also denote by A1 ⊗A2 ⊗... the σ-algebra generated by any ﬁnite family A1, A2, ... of σ-algebras. We shall deal with measurable multi-valued mappings, see e.g. [14, Sect. III.2], [28, Sect. 8.1]. With some abuse, sometimes we shall use the same notation for a multi-valued mapping as well as for any of its selections. Proposition 4.1. Let Ω be a Lipschitz domain of RN , 2 ≤ p < +∞, and set p0 = p/(p − 1). Let F be a family of L(Ω)⊗B(R)⊗B(RN )-measurable mappings Ω×R×RN → P(RN ) such that ∀f ∈ F, for a.e. x, ∀v ∈ R, ∀ξ ∈ RN , f(x, v, ξ) is closed and nonempty, (4.1) ∀f ∈ F, ∀v ∈ R, for a.e. x, the mapping (4.2) RN → P(RN ): ξ 7→ f(x, v, ξ) is monotone, ∀ξ ∈ Lp(Ω)N , ∀M > 0, S p0 N (4.3) {f(·, v, ξ): f ∈ F, kvk 1,p ≤ M} is relatively compact in L (Ω) . W0 (Ω) Setting

0 1,p αf (v) := −∇·f(·, v, ∇v)(⊂ D (Ω)) ∀v ∈ W0 (Ω), ∀f ∈ F, (4.4) 1,p −1,p0 then αf : W0 (Ω) → P(W (Ω)), and the family A = {αf : f ∈ F} is a.l.s.c. in the sense of (3.1). Proof. Let us ﬁx any f ∈ F. By (4.1) and by classical results (see e.g. [14, Sect. III.2], [28, Sect. 8.1]), the mapping (x, v, ξ) 7→ f(x, v, ξ) has an L(Ω)⊗B(R)⊗B(RN )- 1,p 2 measurable selection. For any (v, u) ∈ W0 (Ω) thus the function Ω → P(RN ): x 7→ f(x, v(x), ∇u(x)) has an L(Ω)-measurable selection. (4.5)

1,p Let us set V := W0 (Ω), 0 2 Af (v, u) := −∇·f(·, v, ∇u)(⊂ D (Ω)) ∀(v, u) ∈ V , (4.6) 2 0 and notice that Af : V → P(V ) by (4.3). Let us next check (3.12). For any u ∈ V S and any M > 0, by (4.3) the set Xu = {f(·, v, ∇u): f ∈ F, v ∈ V, kvkV ≤ M} p0 N is relatively compact in L (Ω) . The set {−∇ · η : η ∈ Xu} is then relatively compact in V 0; (3.12) thus holds, and (3.9) follows. Proposition 3.2 then yields the thesis. 2774 AUGUSTO VISINTIN

The next statement stems from Proposition 3.1.

Corollary 4.2. Let the assumptions of Proposition 4.1 be fulfilled, and define αf as in (4.4) for any f ∈ F. For any bounded maximal monotone operator β : V → 0 P(V ), the family {αf + β : f ∈ F} is then a.l.s.c. in the sense of (3.1). The condition (4.3) is a crucial hypothesis of Proposition 4.1. Next we provide sufficient conditions for it to hold within the class of single-valued operators. Henceforth all functions defined on Ω will be extended to RN with vanishing value. Corollary 4.3. Let Ω be a bounded Lipschitz domain of RN , 2 ≤ p < +∞, and set p0 := p/(p − 1). Let F be a family of L(Ω)⊗B(R×RN )-measurable mappings Ω×R×RN → RN , such that ∀f ∈ F, for a.e. x ∈ Ω, ∀v ∈ R, (4.7) RN → RN : ξ 7→ f(x, v, ξ) is continuous and monotone, N ∃C1,C2 > 0 : ∀f ∈ F, for a.e. x ∈ Ω, ∀v ∈ R, ∀ξ ∈ R , p−1 (4.8) |f(x, v, ξ)| ≤ C1|ξ| + C2, 1 0 ∞ N N ∃ϕ ∈ L (Ω), ∃ψ ∈ C (R) ∩ L (R): ∀x1, x2 ∈ R , ∀v1, v2 ∈ R, ∀ξ1, ξ2 ∈ R , p0 f(x1, v1, ξ1) − f(x2, v2, ξ2) (4.9) p p ≤ |ϕ(x1) − ϕ(x2)| + [ψ(v1 − v2) + ψ(|ξ1 − ξ2|)] (|ξ1| + |ξ2| ).

The family A = {αf : f ∈ F} (cf. (4.4)) is then a.l.s.c. in the sense of (3.1). Proof. Let us ﬁrst recall that ξ(· + h) − ξ → 0 in measure as h → 0, ∀ξ ∈ Lp(Ω)N , 1,p N N (4.10) kv(· + h) − vkLp(RN ) ≤ |h|k∇vkLp(RN )N ∀v ∈ W (R ), ∀h ∈ R , see e.g. [11, p. 267]. Denoting by µ the ordinary Lebesgue measure of RN , by the inequality above and the classical Markov inequality, for any ε > 0, sup µx ∈ RN : |v(x + h) − v(x)| > ε → 0 as h → 0. (4.11)

kvkW 1,p(Ω)≤M By (4.9) and by dominated convergence, we then infer that as h → 0

Z 0 p f(x + h, v(x + h), ξ(x + h)) − f(x, v(x), ξ(x)) dx Ω Z (4.12) ≤ |ϕ(x + h) − ϕ(x)| + [ψ(v(x + h) − v(x)) Ω + ψ(|ξ(x + h) − ξ(x)|)] (|ξ(x + h)|p + |ξ(x)|p) dx → 0, 1,p uniformly w.r.t. f ∈ F and w.r.t. all v ∈ W (Ω) such that kvkW 1,p(Ω) ≤ M, for any M > 0. By the classical Kolmogorov-Riesz-Fréchet compactness theorem, we then get (4.3). A priori the family of functions F and the associated family of operators A need not be relatively compact; actually, we did not even equip either family with any topology. The issue of compactness will be addressed in Sects. 5 and 6. This question is also associated to the identification of the limit operator. The next result provides an answer in a rather special case. Proposition 4.4. Let K be a family of equi-bounded and equicontinuous functions R → R+ that is closed w.r.t. uniform convergence. Let Φ be a family of L(Ω)⊗ B(RN )-measurable mappings of the form (x, ξ) 7→ φ(x, ξ) that fulfill (4.7)–(4.9) (here without dependence on v). Let Φ be also closed w.r.t. pointwise convergence. PSEUDO-MONOTONE EQUATIONS 2775

Let {kn} and {φn} be two sequences in K and Φ. Then, for any sequence {un} 1,p p N in W0 (Ω) (1 < p < +∞) and any ξ ∈ L (Ω) , 0 u → u in W 1,p(Ω), k (u )φ (·, ξ) → z in Lp (Ω)N n 0 n n n (4.13) ⇒ ∃k ∈ K, ∃φ ∈ Φ: z = k(u)φ(·, ξ) a.e. in Ω. Proof. It suffices to apply the classical Ascoli-Arzelàand Kolmogorov-Riesz-Fréchet compactess theorems. Uniformly Strict Asymptotic Pseudo-Monotonicity. The next statement follows from Theorem 3.3 and Corollary 4.3. Corollary 4.5. Let F be a family of mappings as in Corollary 4.3. Assume that 1,p there exists a mapping γ that fulfills (2.19)–(2.21) for V := W0 (Ω), and such that ∀f ∈ F, ∀v ∈ V, for a.e. x, the mapping (4.14) RN → P(RN ): ξ 7→ f(x, v, ξ) − γ0(ξ) is monotone.

Then the family A of the αf defined as in (4.4) is uniformly strictly a.l.s.c. in the sense of (3.13). We would like to provide a dual formulation of this result, for the family of inverse operators {α−1 : α ∈ A}. But this seems to encounter some difficulties, since the generalized function −∇·f(·, v, ∇v) does not determine f(·, v, ∇v). We then deal with the mapping v 7→ f(·, v, ∇v), instead of the operator v 7→ −∇·f(·, v, ∇v). First, for any f ∈ F and any v ∈ V let us assume that the mapping RN → RN : ξ 7→ f(·, v, ξ) is invertible a.e. in Ω, and denote its inverse by η 7→ g(·, v, η). Thus g :Ω×R×RN → RN and is L(Ω)⊗B(R×RN )-measurable. Let us denote by G the family of these inverse mappings as f ranges in F. The next result may be regarded as a dual formulation of Theorem 3.3 (for operators of the form (4.6)). Theorem 4.6. Let F be a family of mappings as in Corollary 4.3, G be defined as above and such that

∃C1,C2 > 0 : ∀g ∈ G, for a.e. x, ∀(v, η), 1/(p−1) (4.15) |g(x, v, η)| ≤ C1|η| + C2. 1,p Assume that there exists a function ψ such that, setting V := W0 (Ω) and Hp0 := 0 Lp (Ω), N ψ :(H 0 ) → R is convex and Gˆateaux-diﬀerentiable p (4.16) (with derivative ψ0), N ∀ sequence {θn} in (Hp0 ) , N N (4.17) θn * θ in (Hp0 ) , ψ(θn) → ψ(θ) ⇒ θn → θ in (Hp0 ) ,

∀ sequence {fn} in F, ∀ sequence {un} in V,

setting Fn :=fn(·, un, ∇un), ∀n, Z (4.18) N 0 Fn *F in (Hp0 ) ⇒ lim inf (Fn − F )·[∇un − ψ (Fn)] dx ≥ 0. n→∞ Ω Then

∀ sequence {un} in V , setting Fn := fn(·, un, ∇un) a.e. in Ω, ∀n, Z N Fn *F in (Hp0 ) , lim sup (Fn − F )·∇un dx ≤ 0 (4.19) n→∞ Ω N ⇒ Fn → F in (Hp0 ) . 2776 AUGUSTO VISINTIN

The assumption (4.18) may be regarded as a dual formulation of the uniformly strict asymptotic lower semicontinuity of Sect. 3, see (3.13). In (4.18) G and ψ respectively play the role of F and γ in (3.13). Proof. This argument is essentially a dual version of part (ii) of the proof of Theo- rem 3.3. We have (4.16) Z 0 lim sup ψ(Fn) − ψ(F ) ≤ lim sup (Fn − F )·ψ (Fn) dx n→∞ n→∞ Z Ω 0 ≤ lim sup (Fn − F )·[ψ (Fn) − ∇un] dx n→∞ ΩZ (4.20) + lim sup (Fn − F )·∇un dx n→∞ Z Ω Z 0 = − lim inf (Fn − F )·[∇un − ψ (Fn)] dx + lim sup (Fn − F )·∇un dx. n→∞ Ω n→∞ Ω R If we assume that lim supn→∞ Ω(Fn − F )·∇un dx ≤ 0, then by (4.18) and (4.20) we get

lim sup ψ(Fn) ≤ ψ(F ). n→∞ N On the other hand, Fn *F in (Hp0 ) entails that lim infn→∞ ψ(Fn) ≥ ψ(F ). N Thus ψ(Fn) → ψ(F ), and by (4.17) we conclude that Fn → F in (Hp0 ) . It is promptly seen that Proposition 4.4 and Theorem 4.6 entail the next state- 1,p p0 ment, where we still set V := W0 (Ω) and Hp0 := L (Ω). Corollary 4.7. Let F be a family of mappings as in Corollary 4.3, and let two mappings γ and ψ fulﬁll (2.19)–(2.21), (4.14)–(4.18). Then

∀ sequence {un} in V , setting Fn := fn(·, un, ∇un) a.e. in Ω, ∀n, Z Z N un * u in V,Fn *F in (Hp0 ) , lim sup Fn ·∇un dx ≤ F ·∇u dx (4.21) n→∞ Ω Ω N ⇒ un → u in V,Fn → F in (Hp0 ) .

Remark. Let us consider the family of operators of the form αf (v) := −∇ · f(·, v, ∇v) (cf. (4.4)), or the analogous family with curls, cf. (1.3). Those operators fulfill both hypothesis (4.16) and (4.20) only if the corresponding monotone functions ξ 7→ f(x, v, ξ) are single-valued and injective, for a.e. x and any v. Although this excludes the occurrence of either horizontal or vertical parts in the graph of the mappings f(x, v, ·) for a.e. x and any v, these graphs may have either horizontal or vertical inflection points. No Lipschitz-continuity is thus assumed either for the preceding function or for its inverse. In Sect. 8 this will allow us to extend known results of G-convergence for pseudo-monotone operators, cf. e.g. [37]. Time-Dependent Functions. Let us fix any p ∈ [2, +∞[ as above, any T > 0, and set 1,p 2 p 2 Vp := W0 (Ω),H := L (Ω), Vp := L (0,T ; Vp), H := L (0,T ; H); (4.22) 0 0 hence Vp ⊂ H = H ⊂ Vp with continuous and dense injections. Let us also fix any s ∈ ]0, 1/2[, set ΩT := Ω×]0,T [, and define the separable and reflexive Banach space p s p 1,p0 0 Ws,p(ΩT ) := L (0,T ; Vp) ∩ H (0,T ; H) ⊂L (0,T ; Vp) ∩ W (0,T ; Vp) . (4.23) PSEUDO-MONOTONE EQUATIONS 2777

This will be the domain of the operators acting on time-dependent functions that we shall be concerned with. In this setup a family B of operators Ws,p(ΩT ) ⊂ Vp → 0 P(Vp) is a.l.s.c. iff ∗ ∀ sequence {βn} in B, ∀ bounded sequence {(un, un) ∈ G(βn)}, Z T ∗ ∗ (4.24) un * u in Ws,p(ΩT ) ⇒ lim inf hun, uni dt ≥ hu , ui. n→∞ 0 0 For any operator α : Vp → P(Vp), let us denote by αb the operator Ws,p(ΩT ) → 0 P(Vp) that is canonically associated with α for a.e. t, that is, 0 [αb(u)](t) = α(u(t)) (⊂ Vp) ∀u ∈ Ws,p(ΩT ), for a.e. t ∈ ]0,T [. (4.25) 0 Remark. Let us consider a family A of operators α : Vp → P(Vp) and the corre- 0 sponding family Ab of time-parameterized operators αb : Ws,p(ΩT ) → P(Vp) defined as in (4.25). The weak convergence in Ws,p(ΩT ) does not entail the same conver- 1,p gence in Vp = W0 (Ω) a.e. in ]0,T [, not even for a subsequence. Therefore, even if the family A is a.l.s.c., there is no apparent reason why the time-parameterized family Ab should fulfill the same property. The next result extends Proposition 4.1 to time-dependent functions. Proposition 4.8. Let Ω be a Lipschitz domain of RN , 2 ≤ p < +∞, and set p0 = p/(p − 1). Let F be a family of L(Ω) ⊗ B(R × RN )-measurable mappings Ω×R×RN → P(RN ), such that (4.1) and (4.2) are fulfilled and p N ∀ξ ∈ L (ΩT ) , ∀M > 0, S p0 N (4.26) {f(·, v, ξ): f ∈ F, kvkWs,p(ΩT ) ≤ M} is relatively compact in L (ΩT ) . Setting 0 αf (v) := −∇·f(·, v, ∇v)(⊂ D (Ω)) ∀v ∈ Vp, ∀f ∈ F, (4.27) 0 then αcf : Ws,p(ΩT ) → P(Vp), and the family Ab = {αcf : f ∈ F} is a.l.s.c. in the sense of (4.24). 2 Proof. Let us fix any f ∈ F. As we saw for (4.5), for any (v, u) ∈ Ws,p(ΩT ) , Ω → P(RN ):(x, t) 7→ f(x, v(x, t), ∇u(x, t)) T (4.28) has an L(ΩT )-measurable selection. Let us set 0 2 Acf (v, u) := −∇·f(·, v, ∇u)(⊂ D (ΩT )) ∀(v, u) ∈ Ws,p(ΩT ) , (4.29) 2 0 and notice that Acf : Ws,p(ΩT ) → P(Vp) by (4.25). We claim that

∀M > 0, ∀u ∈ Ws,p(ΩT ), 0 (4.30) {Acf (v, u): f ∈ F, kvkWs,p(ΩT ) ≤ M} is relatively compact in Vp.

For any u ∈ Ws,p(ΩT ), by (4.26) the set Xu = {f(·, v, ∇u): f ∈ F, kvkWs,p(ΩT ) ≤ p0 N M} is relatively compact in L (ΩT ) . The set {−∇·η : η ∈ Xu} is then relatively p0 0 0 compact in L (0,T ; Vp) = Vp. The property (4.30) is thus established. By (4.1), (4.2) and (4.30), the hypotheses of Proposition 3.2 with Vp in place of V are thus fulfilled, and the thesis follows. As we just saw for Proposition 4.1, the other results of this section may also be extended to families of operators of the form (2.12) that depend on the parameter 1,p t: freely speaking, it suffices to replace V = W0 (Ω) by Ws,p(ΩT ) and to use quite 2778 AUGUSTO VISINTIN similar arguments. In particular, by mimicking Corollary 4.3 one may provide a sufficient condition for the crucial condition (4.26). Here we reformulate an extension of Corollary 4.7, that we shall apply ahead. Corollary 4.9. Let F be a family of mappings as in Corollary 4.3, and let two mapping γ and ψ fulfill (2.19)–(2.21), (4.14)–(4.18), with Vp and Hp0 respectively p 1,p p0 replaced by L (0,T ; W0 (Ω)) and L (ΩT ). Then

∀ sequence {un} in Ws,p(ΩT ), setting Fn := fn(·, un, ∇un) a.e. in Ω, ∀n,  p 1,p p0 un * u in L (0,T ; W0 (Ω)),Fn *F in L (ΩT ), ZZ ZZ (4.31) lim sup Fn ·∇un dxdt ≤ F ·∇u dxdt n→∞ ΩT ΩT p 1,p p0 ⇒ un → u in L (0,T ; W0 (Ω)),Fn → F in L (ΩT ).

5. Gamma-compact sequences of functionals. In this section we introduce two nonlinear topologies of weak-type, and prove that equi-coercive sequences of functions V × V 0 → R ∪ {+∞} are Γ-compact with respect to these topologies. In this presentation we partially follow the lines of [47, Chap. 8], which in turn is based on De Giorgi’s theory of Γ-convergence (see e.g. [3,7,8, 17, 18]) and on [2]. The main result is Theorem 5.5; here we need to extend the argument of [47]: that applied to a metrizable topology (denoted by πe), whereas it is not clear whether the topology we deal with here (denoted by πe+) is metrizable. We shall assume throughout that V is a real Banach space, and deﬁne the following linear topologies: (i) we denote by ω the product of the weak topology of V by the weak star topology of V 0, (ii) we denote by ws the product of the weak topology of V by the strong topology of V 0, (iii) we denote by sw∗ the product of the strong topology of V by the weak star topology of V 0, (iv) we denote by s the strong topology of V ×V 0. We also deﬁne two nonlinear topologies, in which the duality pairing will play a key role. First we set π(v, v∗) := hv∗, vi ∀(v, v∗) ∈ V ×V 0. (5.1)

0 We shall denote by πe (πe+, resp.) the coarsest among the topologies of V ×V that are finer than ω and such that the mapping π is continuous (upper semicontinuous, ∗ 0 resp.). For any sequence {(vn, vn)} in V ×V , thus ∗ ∗ 0 (vn, vn) →π˜ (v, v ) in V ×V ⇔ ∗ ∗ ∗ 0 ∗ ∗ (5.2) vn * v in V, vn * v in V , hvn, vni → hv , vi, ∗ ∗ 0 (vn, v ) →(v, v ) in V ×V ⇔ n π˜+ ∗ ∗ ∗ 0 ∗ ∗ (5.3) vn * v in V, vn * v in V , lim sup hvn, vni ≤ hv , vi, and similarly for any net. Thus ω ⊂ πe+ ⊂ πe ⊂ s. We shall say that a topology σ on a Banach space B is boundedly metrizable (boundedly-first-countable, resp.) whenever there exists a topology τ on B that is metrizable (first countable, 4 resp.), and such that τ and σ have the same restriction

4 that is, any point has a countable basis of neighborhoods. PSEUDO-MONOTONE EQUATIONS 2779 to any norm-bounded subset. Note that any boundedly metrizable topology is boundedly-first-countable, but not conversely. 0 Proposition 5.1. [47] If V is separable, then the topologies ω, πe, ws and sw∗ are boundedly metrizable, and the topology πe+ is boundedly-first-countable. Proof. As V 0 is separable, the same holds for V . The space V equipped with the weak topology is then boundedly metrizable, and the same holds for V 0 equipped with the weak star topology; see e.g. [11, Sect. III.6]. Therefore (V ×V 0, ω) (namely, V ×V 0 equipped with the topology ω) is also boundedly metrizable. The same holds for the topologies ws and sw∗. We are left with the proof of the boundedly-first- countability of the topology πe+. Let us denote by R(−) the reals equipped with the topology generated by the open sub-basis {] − ∞, x[: x ∈ R}. For any real sequence {rn} and any r ∈ R,

rn → r in R(−) ⇔ lim sup rn ≤ r (5.4) (this upper limit is meant with respect to the ordinary topology of R!); the limit in R(−) is thus not unique. Because of the density of Q in R, this topology is first-countable; but of course it is not of Hausdorff. By (5.4), for any topological space A, a function A → R is continuous with respect to the topology of R(−) if and only if it is upper semicontinuous with respect to the ordinary topology. By ∗ 0 (5.4), for any sequence {(un, un)} in V ×V , ((u , u∗ ) * (u, u∗) in V ×V 0 ∗ ∗ n n (un, u ) →(u, u ) ⇔ (5.5) n π˜+ ∗ ∗ hun, uni → hu , ui in R(−). 0 0 Let us equip (V ×V , ω)×R(−) with the product topology. As we assumed V to be separable, V has the same property. Any bounded subset of V equipped with the weak topology is then metrizable, hence boundedly-first-countable. The same holds for any bounded subset of V 0 equipped with the weak star topology; see e.g. 0 [11, Sect. III.6]. As R(−) is first-countable, the product space (V ×V , ω)×R(−) is then boundedly-first-countable. Next let us define the mapping θ : V ×V 0 → V ×V 0 ×R :(v, v∗) 7→ (v, v∗, hv∗, vi). (5.6) 0 0 By (5.5), (V ×V , πe+) is homeomorphic to the image set θ(V ×V ) equipped with 0 the topology induced by (V ×V , ω)×R(−). As this image set is boundedly-first- 0 countable, we conclude that the same holds for (V ×V , πe+). (To this author it is not clear whether the topology πe+ is boundedly metrizable. However in this work this property is surrogated by the boundedly-first- countability.) We remind the reader that, for functions defined on a topological space, the definition of Γ-convergence involves the filter of the neighborhoods of each point; see e.g. [3, p. 25-27], [17, p. 38]. If the space is first-countable, that notion may equivalently be reformulated in terms of the family of converging sequences, but this does not apply in general; see e.g. [3, p. 270], [17, Chap. 8]. We shall refer to these two notions as topological and sequential Γ-convergence respectively (we shall also use the notation top-Γ lim and seq-Γ lim). Whenever not otherwise specified, reference to the topological notion should be understood. 0 For instance, a sequence {ψn} of functions V ×V → R ∪ {+∞} sequentially Γ-converges to a function ψ with respect to a topology τ of V ×V 0 if and only if 2780 AUGUSTO VISINTIN

(see e.g. [3, p. 270], [17, p. 86-87]) ∗ 0 ∗ 0 ∀(v, v ) ∈ V ×V , ∀ sequence {(vn, vn)} in V ×V , ∗ ∗ 0 ∗ ∗ (5.7) (vn, vn) →τ (v, v ) in V ×V ⇒ lim infn→∞ ψn(vn, vn) ≥ ψ(v, v ), ∗ 0 ∗ 0 ∀(v, v ) ∈ V ×V , ∃ sequence {(vn, vn)} in V ×V : ∗ ∗ ∗ ∗ (5.8) (vn, vn) →τ (v, v ) and lim supn→∞ ψn(vn, vn) ≤ ψ(v, v ). We shall denote by Γτ lim the Γ-limit with respect to a topology τ, and deﬁne similarly Γτ lim inf and Γτ lim sup. In passing notice that (5.7) ⇔ seq-Γτ lim inf ψ ≥ ψ, n→∞ n (5.9) (5.8) ⇔ seq-Γτ lim supn→∞ ψn ≤ ψ. 0 0 Lemma 5.2. Let V be separable, and two topologies τ1, τ2 on V×V have the same restriction to any bounded subset of this space. Then a sequence {ψn} of functions 0 V ×V → R ∪ {+∞} sequentially Γτ1-converges if and only if it sequentially Γτ2- converges, and the two limits coincide. Proof. By the characterization (5.7) and (5.8), the sequential Γ-convergence only depends on the convergent sequences of V ×V 0. As these sequences are bounded, the sequential Γ-convergence only depends on the restriction of the topology to the bounded subsets of V ×V 0. This yields the above statement. 0 Proposition 5.3. Let V be separable, and τ be any of the topologies ω, πe, ws and 0 0 sw∗ (but not πe+) of V×V . Let σ be a metrizable topology on V×V that is equivalent to τ on any bounded subset of V ×V 0 (by Proposition 5.1, such a topology exists). 0 Let {ψn} be a sequence of functions V ×V → R ∪ {+∞} that is equi-coercive, in the sense that ∗ ∗ 0 ∗ ∀C ∈ R, sup kvkV + kv kV 0 :(v, v ) ∈ V ×V , ψn(v, v ) ≤ C < +∞. (5.10) n∈N Then Γ ψn → ψ topologically w.r.t. τ ⇔ Γ ψn → ψ sequentially w.r.t. σ ⇔ (5.11) Γ ψn → ψ sequentially w.r.t. τ. Proof. For the weak topology this statement is proved in [17, p. 93] after [2, p. 353]; see also [3, p. 285]. That argument is based on the bounded-metrizability of the weak topology. By Proposition 5.1, it may then be repeated verbatim for any of the topologies ω, πe, ws and sw∗ (but not πe+). Next we recall a classical result of Γ-compactness. Lemma 5.4. [3, p. 152], [17, p. 90] If a topological space X has a countable basis, then every sequence {fn} of functions X → R ∪ {±∞} (−∞ included!) has a Γ-convergent subsequence. We are now able to prove a result of Γ-compactness for the weak-type topologies that we introduced above. 0 Theorem 5.5. Let V be separable, τ be any of the topologies ω, πe and πe+ of 0 0 V ×V , and {ψn} be an equi-coercive sequence of functions V ×V → R ∪ {+∞}, in the sense of (5.10). Then, up to extracting a subsequence, ψn topologically and sequentially Γτ-converges to some function ψ, which does not attain the value −∞.

Proof. For the topologies ω, πe (and also ws, sw∗ and s) this result was already proved in Theorem 5.4 of [47]. For the metric topology s it directly stems from Lemma 5.4. Here we prove this result for the (possibly nonmetrizable) topology PSEUDO-MONOTONE EQUATIONS 2781

πe+, which is the result of interest for the remainder of this work. We do so by extending the argument of [47], which in turn is based upon Corollary 8.12 of [17, p. 95]. By Proposition 5.1, the space V ×V 0 may be equipped with a metrizable topology σ, that is equivalent to ω on any bounded subset of V ×V 0. Here we show that 0 (V ×V , σ) has a countable basis. Let us denote by Bn the ball of this space with center the origin and radius n ∈ N. By the assumption of separability of V 0, V 0 and the product space (V ×V , s) are separable; the same then holds for (Bn, s) for any n. As the topology ω is coarser than s,(Bn, σ) = (Bn, ω) is also separable for any n. Thus (V ×V 0, σ) is a countable union of separable subspaces, and so it is also separable. As any separable metric space has a countable basis, the metrizable space (V ×V 0, σ) has a countable basis. The space R(−) also has a countable basis: e.g. the set of all half-lines {] − ∞, r[: 0 r ∈ Q}. The product space (V ×V , σ)×R(−) then also has a countable basis. In the 0 0 proof of Proposition 5.1, we saw that the mapping θ : V ×V → (V ×V , ω)×R(−) 0 (cf. (5.6)) induces onto V ×V the topology πe+. As σ has the same restriction as ω to any norm-bounded subset of V ×V 0, the mapping θ then induces onto V ×V 0 a topology ρ such that ρ and π have the same restriction e+ (5.12) to any norm-bounded subset of V ×V 0.

0 −1 Moreover, if {Bn} is a countable basis of (V ×V , σ)×R(−), the family {θb (Bn)} is a countable basis of (V ×V 0, ρ). By Lemma 5.4, {ψn} then has a topologically Γρ-convergent subsequence. This is also sequentially Γρ-convergent, since the topology ρ is first-countable. By (5.12) and Lemma 5.2, this subsequence then Γπe+-converges sequentially to a function ψ : X → R ∪ {±∞}. By (5.10), ψ does not attain the value −∞ (although it might be identically +∞). Remarks. (i) In Theorem 5.5 the Γ-limit may be identically +∞. (ii) It is known that a sequence may have two different Γ-limits w.r.t. two different topologies. The weaker is the topology, the smaller is the Γ-limit, if it exists. Next we complete Theorem 5.5 by proving a result of Γ-compactness with respect to the topologies ws, sw∗ and s under weaker assumptions of equi-coerciveness. Loosely speaking, global (bounded, resp.) equi-coerciveness is here assumed with respect to a variable that corresponds to the weak or weak star (strong, resp.) topology. 0 Proposition 5.6. Let V be separable, and {ψn} be a sequence of functions V × V 0 → R ∪ {+∞}. Then: 0 (i) If {ψn} is equi-bounded from below by some function V ×V → R ∪ {+∞}, 0 then ψn topologically and sequentially Γ-converges to some function ψe : V ×V → R ∪ {+∞} with respect to the topology s, up to extracting a subsequence. (ii) If {ψn} is equi-coercive with respect to the first variable, boundedly with respect to the second one, in the sense that

∗ 0 ∗ ∗ ∀C ∈ R, sup kvkV :(v, v ) ∈ V ×V , ψn(v, v ) + kv kV 0 ≤ C < +∞, (5.13) n∈N then ψn topologically and sequentially Γ-converges to some function ψ with respect to the topology ws, up to extracting a subsequence. 2782 AUGUSTO VISINTIN

(iii) Dually, if {ψn} is equi-coercive with respect to the second variable, boundedly with respect to the ﬁrst one, i.e., ∗ ∗ 0 ∗ ∀C ∈ R, sup kv kV 0 :(v, v ) ∈ V ×V , ψn(v, v ) + kvkV ≤ C < +∞, (5.14) n∈N then ψn topologically and sequentially Γ-converges to some function ψe with respect to the topology sw∗, up to extracting a subsequence. Proof. The ﬁrst statement directly follows from Lemma 5.4. ∗ ∗ ∗ In view of proving the second assertion, let us set θ(v ) := kv kV 0 for any v ∈ 0 0 V , and γn := ψn + θ in V × V , for any n. By (5.13), this sequence is equi- coercive in the sense of (5.10). By Theorem 5.5 (actually, Theorem 5.4 of [47]), γn then topologically and sequentially Γ-converges to some function γ with respect to the topology ws, up to extracting a subsequence. As the function θ is strongly 0 continuous on V , it follows that ψn = γn − θ topologically and sequentially Γ- converges to ψ := γ − θ with respect to the topology ws. The argument for the third statement is analogous, and is omitted.

6. Compact sequences of operators. In this section we show two main results: (i) the graphs of any equi-coercive and asymptotically π-lower-semicontinuous (“a.l.s.c.”, for short, cf. (3.1)) sequence of operators are sequentially compact in the sense of Kuratowski with respect to the topology πe+ (Theorem 6.2); (ii) any operator that is a πe+-cluster point of those operators is generalized pseudo-monotone, in the sense of (2.13) (Theorem 6.3). Kuratowski and Graph Convergence. We remind the reader that a sequence {An} of subsets of a topological space (X, τ) sequentially converges in the sense of Kuratowski (or just K-converges) to a set A with respect to τ (we then write either K An → A w.r.t. τ, or A = Kτ limn→∞ An) if and only if

IAn sequentially Γτ-converges to IA. (6.1)

(By IA we still denote the indicator function of any set A.) The corresponding topological convergence is similarly deﬁned; see e.g. [3, p. 94]. We shall be especially concerned with the sequential limit and the corresponding sequential lower- and upper-limits, that we denote by seq-Kτ lim inf and seq-Kτ lim inf, respectively. The Kuratowski convergence induces a notion of convergence for operators: given 0 0 a topology τ on V ×V , we shall say that a sequence {αn} of operators V → P(V ) sequentially converges to α in the sense of the graph with respect to τ (and write G either αn → α with respect to τ, or α = Gτ limn→∞ αn) if

G(αn) sequentially K-converges to G(α) with respect to τ. (6.2) The corresponding lower and upper G-limits are defined analogously. This extends a classical definition, see e.g. [3, p. 360]. The relation between solutions (in V ) and data (in V 0) of a large class of problems may be represented in this way. Note that: (i) α ⊃ Gτ lim supn→∞ αn means that the relation α is fulfilled in the limit. However the set G(α) might be too large; that is, it might include spurious elements, that cannot be represented as the limit of any sequence that fulfills the relations −1 αns. (This is obviously excluded whenever the operator α is single-valued.) (ii) α ⊂ Gτ lim infn→∞ αn means that any element that fulfills α may be represented as the limit of a subsequence that fulfills those relations. However G(α) might be too small, that is, it might miss some asymptotic elements. PSEUDO-MONOTONE EQUATIONS 2783

G Whenever αn → α with respect to a topology τ, no limit element is missing in G(α), nor any spurious one is included. One may thus regard α as a genuine G representative of the asymptotic behavior of the sequence {αn} only if αn → α. In several cases proving this convergence may be nontrivial: this may explain why in several works one is satisfied with just proving the property (i). Operator Compactness. The compactness Theorems 5.5 and 5.6 may be applied 0 to the Kuratowski convergence of subsets of V ×V . But a sequence {IG(αn)} is equi-coercive only if the sequence of graphs {G(αn)} is equi-bounded, and this even excludes any maximal monotone operator. Next we overcome this difficulty by assuming that the sequence of operators {αn} (rather than the sequence of mappings {IG(αn)}) is equi-coercive, and also a.l.s.c. in the sense of (3.1). 0 Proposition 6.1. Let V be separable, and τ be any of the topologies πe, ws, sw∗, 0 0 s (but neither πe+ nor ω) of V ×V . Let A be a family of operators V → P(V ) such that ∗ ∗ ∗ ∀C ∈ R, sup kvkV + kv kV 0 :(v, v ) ∈ G(α), hv , vi ≤ C < +∞. (6.3) α∈A 0 Then there exist an operator α : V → P(V ) and a sequence {αn} in A such that, defining π as in (5.1), I + π →Γ I + π G(αn) G(α) (6.4) topologically and sequentially w.r.t. the topology τ,

I →Γ I (i.e., α G→ α) G(αn) G(α) n (6.5) topologically and sequentially w.r.t. the topology τ. (In (6.4), π might also be replaced by g ◦ π for any continuous and coercive real function g.)

Proof. By (6.3), the family {ψ := IG(α) + π : α ∈ A} is equi-coercive in the sense of (5.10). Either by Theorem 5.5 or Proposition 5.6 (this depends on the selected topology τ), then there exist a function ψ : V ×V 0 → R ∪ {+∞} and a sequence {αn} in A such that ψ = I + π →Γ ψ n G(αn) (6.6) topologically and sequentially w.r.t. the topology τ. As the function π is obviously continuous with respect to the topology τ, it follows that I = ψ − π →Γ ψ − π G(αn) n (6.7) topologically and sequentially w.r.t. the topology τ. On the other hand it is promptly seen that (whenever existing) the Γ-limit of any sequence of indicator functions is necessarily an indicator function. Therefore 0 (6.5) holds for some operator α : V → P(V ). By (6.5) and (6.7) ψ − π = IG(α), and (6.6) yields (6.4). Next we show that the asymptotic lower-semicontinuity provides the Γ-compactness with respect to the topology πe+, which was excluded in Proposition 6.1. Theorem 6.2. Let V 0 be separable and A be an a.l.s.c. family of operators (in the 0 sense of (3.1)) V → P(V ); that is, for any sequence {αn} in A, ∗ 0 ∗ ∀(v, v ) ∈ V ×V , ∀ sequence {(vn, vn) ∈ G(α`n )}, ∗ ∗ ∗ ∗ (6.8) (vn, v ) →(v, v ) ⇒ (vn, v ) → (v, v ). n π˜+ n π˜ 2784 AUGUSTO VISINTIN

If A fulﬁlls (6.3), then there exist an operator α : V → P(V 0) and a sequence {αn} in A such that I + π →Γ I + π G(αn) G(α) (6.9) topologically and sequentially w.r.t. the topology πe+, I →Γ I (i.e., α G→ α) G(αn) G(α) n (6.10) sequentially w.r.t. the topology πe+. (In (6.9), π might be replaced by g ◦ π for any continuous and coercive real function g.) Proof. A priori it is not obvious that (6.9) is equivalent to (6.10), since the mapping π is not continuous with respect to the topology πe+. We split this argument into two steps.

(i) By (6.3) the sequence {ψn := IG(αn) +π} is equi-coercive in the sense of (5.10). By Theorem 5.5, then there exists a function ψ : V ×V 0 → R ∪ {+∞} such that, up to extracting a subsequence, Γ ψn → ψ topologically and sequentially w.r.t. the topology πe+. (6.11) We claim that then I = ψ − π →Γ ψ − π sequentially w.r.t. the topology π . (6.12) G(αn) n e+ As the Γ-limit of any sequence of indicator functions (if existing) is an indicator function, this will entail (6.9) and (6.10) for some operator α : V → P(V 0). (ii) In order to prove (6.12), by the definition of sequential Γ-limit (see (5.7) and (5.8)) and by (6.11), it suffices to check the two following properties: ∗ ∗ 0 ∗ ∗ ∀{(vn, v )}, ∀(v, v ) in V ×V , (vn, v ) →(v, v ) ⇒ n n π˜+ ∗ ∗ ∗ ∗ (6.13) lim infn→+∞ [ψn(vn, vn) − hvn, vni] = lim infn→+∞ ψn(vn, vn) − hv , vi, ∗ ∗ 0 ∗ ∗ ∀{(vn, v )}, ∀(v, v ) in V ×V , (vn, v ) →(v, v ) ⇒ n n π˜+ ∗ ∗ ψn(vn, vn) → ψ(v, v ) if and only if (6.14) ∗ ∗ ∗ ∗ ∗ IG(αn)(vn, vn) = ψn(vn, vn) − hvn, vni → ψ(v, v ) − hv , vi ∗ 0 (these limits need not be finite). Let us assume that a sequence {(vn, vn)} in V ×V ∗ ∗ is such that (vn, v ) →(v, v ). First notice that, by the boundedness of the sequence n π˜+ ∗ {hvn, vni}, either of the inferior limits of (6.13) is infinite if and only if so is the other one. If the left side of the equality of (6.13) is finite, then there exists a 0 ∗ 0 subsequence, that we index by n , such that (vn0 , vn0 ) ∈ G(αn0 ) for any n . By (6.8) ∗ ∗ then hvn0 , vn0 i → hv , vi, whence ∗ ∗ ∗ ∗ lim inf [ψn0 (vn0 , vn0 ) − hvn0 , vn0 i] = lim inf ψn0 (vn0 , vn0 ) − hv , vi. (6.15) n0→+∞ n0→+∞ ∗ ∗ As this also holds for any subsequence {(vn0 , vn0 ) ∈ G(αn0 )} of {(vn, vn)}, (6.13) then follows. ∗ ∗ Next we prove (6.14). If either ψn(vn, vn) or IG(αn)(vn, vn) has a finite limit, ∗ ∗ ∗ then (vn, vn) ∈ G(αn) for any sufficiently large n, so that by (6.8) hvn, vni → hv , vi. Therefore ∗ ∗ ∗ limn→+∞ IG(αn)(vn, vn) = limn→+∞ [ψn(vn, vn) − hvn, vni] ∗ ∗ = limn→+∞ ψn(vn, vn) −limn→+∞hvn, vni (6.16) ∗ ∗ = limn→+∞ ψn(vn, vn) − hv , vi. (6.14) is thus established, and with it the convergence (6.12). PSEUDO-MONOTONE EQUATIONS 2785

Theorem 6.3. Let V be reflexive and separable and {αn} be a sequence of operators V → P(V 0) such that I →Γ I (i.e., α G→ α) sequentially w.r.t. the topology π . (6.17) G(αn) G(α) n e+ Then α is generalized pseudo-monotone, in the sense of (2.13). Proof. It is known that any Γ-limit is lower semicontinuous, see e.g. [17, p. 57]. By (6.17) the function IG(α) + π is thus sequentially lower semicontinuous with respect ∗ 0 to the topology πe+. For any sequence {(vm, vm)} in V ×V such that ∗ ∗ 0 (vm, v ) →(v, v ) in V ×V , (6.18) m π˜+ then ∗ ∗ ∗ ∗ lim inf [IG(α)(vm, vm) + hvm, vmi] ≥ IG(α)(v, v ) + hv , vi. m→∞ ∗ ∗ ∗ If (vm, vm) ∈ G(α) for any sufficiently large m, then (v, v ) ∈ G(α) (i.e., v ∈ α(v)), ∗ ∗ and the preceding inequality reads lim infm→∞ hvm, vmi ≥ hv , vi. On the other ∗ ∗ ∗ ∗ hand, by (6.18) lim supm→∞ hvm, vmi ≤ hv , vi. Hence hvm, vmi → hv , vi, and ∗ ∗ thus (vm, vm) →π˜ (v, v ). The generalized pseudo-monotonicity property (2.13) is thus established. The next statement gathers the main conclusions of this section. Corollary 6.4. Let V be reflexive and separable. Let A be a family of operators V → P(V 0), that is a.l.s.c. in the sense of (3.1) and fulfills (6.3). Then there exist 0 an operator α : V → P(V ) and a sequence {αn} in A such that I →Γ I (i.e., α G→ α) sequentially w.r.t. the topology π . (6.19) G(αn) G(α) n e+ This entails that α is generalized pseudo-monotone, in the sense of (2.13). If the family A is uniformly strictly a.l.s.c. in the sense of (3.13), then α is strictly generalized pseudo-monotone, in the sense of (2.19)–(2.22). Proof. Theorem 6.2 yields (6.10). By Theorem 6.3 α then fulfills (2.13). The final statement is a straightforward consequence of (2.22). It should be noticed that in this section the approximating operators have not been assumed to be (generalized) pseudo-monotone. Remark. Definitions and results of Sects. 5 and 6 take over to operators that act on time-dependent functions, by replacing the space V by V := L2(0,T ; V ).

7. Doubly-nonlinear parabolic ﬂows: Existence result. In this section we introduce a Cauchy problem for doubly-nonlinear parabolic equations of the form

Dt∂ϕ(u) + α(u) 3 h; (7.1) here ϕ is a convex and lower semicontinuous mapping, and α is a generalized pseudo- monotone operator, in the sense of (2.13). We prove existence of a solution. Due to the double nonlinearity, in general the uniqueness of the solution is far from obvious. In the next section we shall address the structural stability of the Cauchy problem associated to (7.1), w.r.t. variations of the data ϕ, α, h. Functional Set-up. Let Ω be a bounded domain of RN of Lipschitz class, p ∈ [2, +∞[, p0 = p/(p − 1), and set

1,p p 2 Vp := W0 (Ω),Hp := L (Ω),H := L (Ω). (7.2) 2786 AUGUSTO VISINTIN

Thus

Vp,Hp,H are reﬂexive and separable real Banach spaces, 0 0 (7.3) Vp ⊂ Hp ⊂ H = H ⊂ Hp0 ⊂ Vp with continuous and dense injections,

the injection Vp → Hp is compact. (7.4) Let us assume that α, ϕ, h and w0 are prescribed, such that

ϕ : Hp → R is convex (hence continuous), (7.5) ∃C , ..., C > 0 : ∀v ∈ H ,C kvkp + C ≤ ϕ(v) ≤ C kvkp + C , (7.6) 1 4 p 1 Hp 2 3 Hp 4

∃C5 > 0 : ∀(v1, w1), (v2, w2) ∈ G(∂ϕ), 2 (7.7) hw1 − w2, v1 − v2i ≥ C5kv1 − v2kH , 0 α : Vp → P(Vp) is generalized pseudo-monotone (cf. (2.13)), (7.8)

∀v ∈ Vp, the set α(v) is nonempty, closed and convex, (7.9)

∃a, b > 0 : ∀(v, v∗) ∈ G(α), hv∗, vi ≥ akvkp − b, (7.10) Vp ∗ ∗ p0 p ∃c1, c2 > 0 : ∀(v, v ) ∈ G(α), kv k 0 ≤ c1kvk + c2, (7.11) Vp Vp p0 0 0 h ∈ L (0,T ; Vp), w ∈ Hp0 . (7.12) Notice that, denoting by ϕ∗ the convex conjugate function of ϕ, by (7.6)

p0 ∗ p0 ∃C˜1, ..., C˜4 > 0 : ∀v ∈ Hp0 , C˜1kvk + C˜2 ≤ ϕ (v) ≤ C˜3kvk + C˜4. (7.13) Hp0 Hp0

In the sequel, we shall denote the space of weakly continuous maps [0,T ] → Hp 0 by Cw([0,T ]; Hp). We remind the reader that

∞ 1,p0 0 0 L (0,T ; Hp) ∩ W (0,T ; Vp) ⊂ Cw([0,T ]; Hp), (7.14) see e.g. Sect. 3.8.4 of [34]. In the next section we shall deal with the weak structural stability of the following Cauchy problem, for which here we prove an existence result. Problem 7.1. Find

0 1,p0 0 p p0 0 w ∈ Cw([0,T ]; Hp0 ) ∩ W (0,T ; Vp), u ∈ L (0,T ; Vp), z ∈ L (0,T ; Vp), (7.15) such that

0 Dtw + z = h in Vp, a.e. in 0,T [, (7.16)

w ∈ ∂ϕ(u) in Hp0 , a.e. in 0,T [, (7.17) 0 z ∈ α(u) in Vp, a.e. in 0,T [, (7.18) 0 0 w(0) = w in Vp. (7.19) Proposition 7.1. Let (7.2), (7.3), (7.5)–(7.12) be fulﬁlled. Then Problem 7.1 has a solution such that

∞ 1,p0 0 w ∈ L (0,T ; Hp0 ) ∩ W (0,T ; Vp), s p u ∈ H (0,T ; H) ∩ L (0,T ; Vp) ∀s < 1/2, (7.20) p0 0 z ∈ L (0,T ; Vp). PSEUDO-MONOTONE EQUATIONS 2787

Moreover, for any s < 1/2, the norms of these functions in these spaces are bounded by quantities that only depend on the constants that occur in (7.6)–(7.11), 0 0 on khk p 0 , and on kw kH 0 . L (0,T ;Vp ) p Proof. This argument diﬀers from the standard procedure that is used for doubly- nonlinear quasilinear parabolic equations, as the operator α is assumed to be generalized pseudo-monotone (rather than just maximal monotone). For instance, here it does not seem feasible to multiply the time-increment of the approximate equation by the time-increment of the variable u. This obstruction is here overcome via a further estimate procedure, which may be found e.g. in [1, 46]; see part (iv) of this proof. (i) Time-Discrete Approximation. In order to approximate our problem by an implicit time-discretization scheme, let us ﬁx any m ∈ N, set k = T/m and Z `k ` 1 0 hm := h(τ)dτ in Vp, for ` = 1, . . . , m. (7.21) k (`−1)k

` ` 0 ` Problem 7.2m. Find um ∈ Vp, zm ∈ Vp and wm ∈ Hp for ` = 1, . . . , m, such 0 0 that, setting wm = w , w` − w`−1 m m + z` = h` in V 0, for ` = 1, . . . , m, (7.22) k m m p ` ` wm ∈ ∂ϕ(um) in Hp0 , for ` = 1, . . . , m, (7.23) ` ` 0 zm ∈ α(um) in Vp, for ` = 1, . . . , m. (7.24) 0 Deﬁning the operator Bk := ∂ϕ + kα : Vp → P(Vp), this system also reads ` ` `−1 0 Bk(um) 3 khm + wm in Vp, for ` = 1, . . . , m. (7.25)

It is easily checked that Bk fulﬁlls the assumptions of Lemma 2.1. For this inclusion, existence of a solution can then be proved step by step. ` (ii) A Priori Estimates. Let us multiply (7.22) by kum and sum for ` = 1, . . . , n, ` ∗ ` for any n ∈ {1, . . . , m}. The inclusion (7.23) also reads um ∈ ∂ϕ (wm) in Hp for any `, whence n n X ` `−1 ` X ∗ ` ∗ `−1 ∗ n ∗ 0 Hp0 hwm − wm , umiHp ≥ [ϕ (wm) − ϕ (wm )] = ϕ (wm) − ϕ (w ). `=1 `=1 By (7.6)–(7.12), the following estimates then easily follow: m m n X n p X n p0 max kwmkH 0 , k kumkV , k kzmkV 0 n=1,...,m p p p (7.26) n=1 n=1 ≤ Constant (independent of m), whence, by comparison in (7.22),

m 0 X w` − w`−1 p k m m ≤ Constant (independent of m). (7.27) k V 0 n=1 p (iii) Time-Continuous Approximation. Let us set ` wm := piecewise linear time-interpolate of {wm(`k) := wm}n=0,...,m, (7.28) ` w¯m(t) := wm if (` − 1)k < t ≤ `k, for ` = 1, . . . , m, (7.29) 2788 AUGUSTO VISINTIN

¯ and deﬁneu ¯m, z¯m, hm and so on similarly. The system (7.22)–(7.24) thus reads ¯ 0 Dtwm +z ¯m = hm in Vp, a.e. in 0,T [, (7.30)

w¯m ∈ ∂ϕ(¯um) in Hp0 , a.e. in 0,T [, (7.31) 0 z¯m ∈ α(¯um) in Vp, a.e. in 0,T [. (7.32) The a priori estimates (7.26)–(7.27) then read

0 p 0 kwmkL∞(0,T ;H 0 )∩W 1,p (0,T ;V 0), kumkL (0,T ;Vp), kzmkLp (0,T ;V 0) p p p (7.33) ≤ Constant (independent of m).

(iv) A Further a Priori Estimate. First let us set

(τhφ)(t) := φ(t + h) ∀h, t ∈ R, ∀φ : R → R. (7.34)

Next let us set um(t) := um(0) for any t < 0, multiply (7.30) by um − τ−hum, and integrate in ]0,T [. By (7.7) and (7.33), we have Z T

Dtwm, um − τ−hum dt ≤ 2kDtwmk p0 0 kumkLp(0,T ;V ) L (0,T ;Vp ) p (7.35) h ≤ Constant (independent of m, h).

1,p0 0 As the sequence {wm} is bounded in W (0,T ; Vp), we infer that w − τ w σ := D w − m −h m m,h t m h (7.36) p0 0 is bounded in L (0,T ; Vp), uniformly with respect to m, h; moreover, by (7.7) 2 wm − τ−hwm, um − τ−hum ≥ C5kum − τ−humkH a.e. in [0,T ]. (7.37) We thus have Z T 2 (7.37) Z T kum − τ−humkH Dwm − τ−hwm E C5 dt ≤ , um − τ−hum dt h h h h (7.36) Z T

≤ Dtwm, um − τ−hum dt (7.38) h + kσ k 2 0 ku − τ u k 2 m,h L (0,T ;Vp ) m −h m L (0,T ;Vp) ≤ Constant (independent of h, m). 1 Therefore for any ε ∈ ]0, 2 [ 2 kumkH1/2−ε(0,T ;H) ZZ 2 2 kum(t) − um(s)kH = kumkL2(0,T ;H) + 2−2ε dtds ]0,T [2 |t − s| Z T Z t 2 2 kum(t) − um(t − h)kH = kumkL2(0,T ;H) + 2 dt 2−2ε dh (7.39) 0 0 h Z T Z T 2 2 2ε−1 kum(t) − τ−hum(t)kH = kumkL2(0,T ;H) + 2 dh h dt 0 h h (7.33),(7.38) ≤ Constant (independent of m), that is, s the sequence {um} is bounded in H (0,T ; H), for any s < 1/2. (7.40) PSEUDO-MONOTONE EQUATIONS 2789

(v) Limit Procedure. By the estimates (7.33) and (7.40) there exist u, w, z such that, possibly taking m → ∞ along a subsequence, for any s < 1/2

∗ ∞ 1,p0 0 wm * w in L (0,T ; Hp0 ) ∩ W (0,T ; Vp), (7.41) p s u¯m, um * u in L (0,T ; Vp) ∩ H (0,T ; H), (7.42) p0 0 z¯m, zm * z in L (0,T ; Vp). (7.43) By passing to the limit in (7.30) we get (7.16). We are left with the passage to the limit in the nonlinear relations (7.31) and (7.32). The inclusion (7.31) is tantamount to Z T Z T ∗ [ϕ(¯um) + ϕ (w ¯m)] dt ≤ Hp0 hw¯m, u¯miHp dt. (7.44) 0 0 By (7.42) and by the classical Lions-Aubin compactness theorem (see e.g. [40]),

p u¯m, um → u in L (0,T ; Hp), so that by (7.41) Z T

lim Hp0 hw¯m, u¯miHp dt m→∞ 0 Z T Z T (7.45)

= lim Hp0 hwm, u¯miHp dt = Hp0 hw, uiHp dt. m→∞ 0 0 On account of the sequential weak lower semicontinuity of ϕ and ϕ∗, by passing to the inferior limit in (7.44) the inequality is then preserved. This yields (7.17). We are left with the identiﬁcation of z. By (7.33) each wm is continuous as a mapping from ]0,T [ to Hp0 equipped with the weak topology. Possibly extracting a further sequence, by (7.14) we then have

w¯m(T ) = wm(T ) * w(T ) in Hp0 , (7.46) whence, by the sequential weak lower semicontinuity of the functional ϕ∗,

∗ ∗ lim inf ϕ (wm(T )) ≥ ϕ (w(T )). (7.47) m→∞ Hence T T Z (7.30) Z lim sup 0 hz¯ , u¯ i dt = lim sup 0 hh¯ − D w , u¯ i dt Vp m m Vp Vp m t m m Vp m→∞ 0 m→∞ 0 Z T (7.31) ∗ ∗ 0 = lim 0 hh¯ , u¯ i dt − lim inf ϕ (w (T )) + ϕ (w ) Vp m m Vp m m→∞ 0 m→∞ Z T (7.48) ∗ ∗ 0 ≤ 0 hh, ui dt − ϕ (w(T )) + ϕ (w ) Vp Vp 0 T T Z (7.16) Z = 0 hh − D w, ui dt = 0 hz, ui dt. Vp t Vp Vp Vp 0 0 By the generalized pseudo-monotonicity of α, (7.18) then follows. This concludes the proof of existence of a solution of Problem 7.1n. The ﬁnal statement of Proposition 7.1 follows from (7.33) and (7.40). 2790 AUGUSTO VISINTIN

8. Weak structural stability of doubly-nonlinear parabolic flows. In this section we address the structural stability of Problem 7.1, as ϕ varies through a family of convex, lower semicontinuous mappings, and α through a family of a.l.s.c. operators, in the sense of (3.1). First we introduce what we name a global-in-time problem, and prove that: (i) any bounded family of operators has a cluster point w.r.t. a suitable topology (see Propositions 8.1 and 8.2), and (ii) the corresponding solutions accumulate at a limit solution (see Theorem 8.3). As a priori this solution might exhibit memory, under further hypotheses we then show that it also solves a natural pointwise-in-time formulation (see Theorem 8.4). Global-in-Time Solution. We shall use the following notation: 0 p 1 p ∈ [2, +∞[, p = p−1 , s ∈ 0, 2 [, 1,p p 2 p0 Vp := W0 (Ω),Hp := L (Ω),H := L (Ω),Hp0 := L (Ω), p s (8.1) Ws,p(ΩT ) := L (0,T ; Vp) ∩ H (0,T ; H) p 1,p0 0 ⊂L (0,T ; Vp) ∩ W (0,T ; Vp) . 0 Let us also define the topologies πe and πe+ in Vp × Vp as we did in Sect. 5, replacing the space V by Vp and the duality pairing h·, ·i by Z T ∗ ∗ ∗ 0 hhv , vii := hv (t), v(t)i dt ∀(v, v ) ∈ Vp ×Vp. 0 Definitions and results of Sects. 5 and 6 take over to operators that act on time- dependent functions, simply by replacing the space V by Vp. For any n, let us denote 0 by αb the time-dependent operator Vp → P(Vp) that is canonically associated with 0 a mapping α : Vp → P(Vp) for a.e. t, as in (4.25). Let us also denote by Φ the R T functional Vp → R : v 7→ 0 ϕ(v) dt. The system (7.16)–(7.18) thus also reads 0 Dt∂Φ(u) + αb(u) 3 h in Vp. (8.2) We shall say that this equation is global-in-time, at variance with the equivalent pointwise-in-time system (7.16)–(7.18). It should be noticed that any αb is causal, in the sense that ∀v , v ∈ W (Ω ), ∀t ∈ ]0,T ], v = v in [0, t] 1 2 s,p T 1 2 (8.3) ⇒ [αb(v1)](t) = [αb(v2)](t), and has no memory, in the sense that

∀v1, v2 ∈ Ws,p(ΩT ), ∀t ∈ ]0,T ], v1(t) = v2(t) ⇒ [αb(v1)](t) = [αb(v2)](t). (8.4) Proposition 8.1. Let us prescribe (8.1), and let ϕ vary through a family Ψ of functions that fulﬁll (7.5) and (7.6). Then there exist a sequence {ϕn} in Ψ and a function ϕ : Hp → R such that Γ ϕn → ϕ in Hp. (8.5) This entails that ∗ Γ ∗ ϕn → ϕ sequentially weakly in Hp0 , (8.6) Z T Z T Γ p ϕn(·) dt → ϕ(·) dt in L (0,T ; Hp), (8.7) 0 0 Z T Z T ∗ Γ ∗ p0 ϕn(·) dt → ϕ (·) dt sequentially weakly in L (0,T ; Hp0 ). (8.8) 0 0 PSEUDO-MONOTONE EQUATIONS 2791

Moreover, the sequences (8.5)–(8.8) also converge pointwise in the respective spaces to the respective Γ-limits. Proof. By Lemma 5.4, (8.5) holds for a suitable sequence. By Proposition 5.12 of [17, p. 50] and (7.5), (7.6), then ϕn → ϕ pointwise in Hp. The same then holds for the corresponding integral functionals. By the result that we have just quoted, these functionals also converge pointwise. By Theorem 3.11 of [3, p. 282], (8.5) and (8.7) respectively entail (8.6) and (8.8), sequentially weakly as well as pointwise.

Proposition 8.2. Let us prescribe (8.1), and let α vary through a family A of mappings that fulﬁll (7.2), (7.3), (7.8)–(7.11). Let us denote by Ab the corresponding 0 family of time-dependent operators Ws,p(ΩT ) ⊂ Vp → P(Vp), and assume that

the family Ab is a.l.s.c. in the sense of (3.1) (here in Vp), (8.9) i.e., deﬁning the πe- and πe+-convergence as in (5.2) and (5.3),

∀ subsequence {αn} in A, ∗ 0 ∗ ∀(v, v ) ∈ Ws,p(ΩT )×Vp, ∀ sequence {(vn, vn) ∈ G(αbn)}, (8.10) ∗ ∗ ∗ ∗ (vn, v ) →(v, v ) ⇒ (vn, v ) → (v, v ). n π˜+ n π˜

Then there exist a sequence {αn} in A and an operator β : Ws,p(ΩT ) ⊂ Vp → 0 P(Vp) such that

G 0 αcn → β sequentially in Vp ×Vp w.r.t. the topology πe+. (8.11) Moreover, the operator β is generalized pseudo-monotone (in the sense of (2.13), here rewritten in Ws,p(ΩT )) and is causal (in the sense of (8.3)). Proof. By (7.10) and (7.11) the equi-coerciveness condition (6.3) is fulﬁlled. By Corollary 6.4 then there exist a sequence {αn} in A and an operator β as in (8.11), and this is generalized pseudo-monotone. We are left with the proof of the causality of β. Let us ﬁx any t¯ ∈ ]0,T [ and ¯ ¯ assume that u1, u2 ∈ Ws,p(ΩT ) coincide in [0, t]. Then αcn(u1) = αcn(u2) in [0, t] for any n. This equality is preserved in the limit, and so β(u1) = β(u2) in [0, t¯].

Next we come to the structural stability of the sequence of Problems 7.1n. Theorem 8.3. Let us prescribe (8.1) and the assumptions of Propositions 8.1 and 8.2. Let the sequences {ϕn} in Ψ, {αn} in A, the mapping ϕ : Hp → R, and the 0 operator β : Ws,p(ΩT ) ⊂ Vp → P(Vp) be as in (8.5) and (8.11). Let

0 p0 0 w ∈ Hp0 , hn * h in L (0,T ; Vp). (8.12)

For any n, let (un, wn, zn) be a solution of the corresponding Problem 7.1n (by Proposition 7.1 this solution exists). Then there exists a triplet (u, w, z) such that, up to extracting subsequences,

∗ ∞ 1,p0 0 wn * w in L (0,T ; Hp0 ) ∩ W (0,T ; Vp), (8.13) p s un * u in L (0,T ; Vp) ∩ H (0,T ; H), (8.14) p0 0 zn * z in L (0,T ; Vp). (8.15) 2792 AUGUSTO VISINTIN

Moreover, this entails that 0 Dtw + z = h in Vp, (8.16)

w ∈ ∂ϕ(u) in Hp0 , a.e. in 0,T [, (8.17) 0 z ∈ β(u) in Vp, (8.18) 0 0 w(0) = w in Vp. (8.19) Proof. By Proposition 7.1, for any s < 1/2

kwnk ∞ 1,p0 0 , kunkLp(0,T ;V )∩Hs(0,T ;H), kznk p0 0 L (0,T ;Hp0 )∩W (0,T ;Vp ) p L (0,T ;Vp ) ≤ Constant (independent of n). Therefore there exists a triplet (u, w, z) that fulﬁlls (8.13)–(8.15). As the equation (7.16) holds for any n, (8.16) follows. In order to prove (8.17), let us notice that the inclusion (7.17) is tantamount to Z T Z T Z T ∗ ϕn(un) dt + ϕn(wn) dt ≤ Hp0 hwn, uniHp dt. (8.20) 0 0 0

By the compactness of the injection Vp → Hp and a classical Schauder’s theorem, 0 the dual injection Hp0 → Vp is also compact. By the classical Aubin-Lions theorem (see e.g. [40]), then p0 0 wn → w in L (0,T ; Vp). Hence Z T Z T

Hp0 hwn, uniHp dt → Hp0 hw, uiHp dt. (8.21) 0 0 On account of (8.7) and (8.8), by passing to the inferior limit in (8.20) the inequality is then preserved. (8.17) thus holds. We are left with the proof of (8.18). By Proposition 8.1, ∗ 0 ∗ 0 ϕn(w ) → ϕ (w ). (8.22)

By (7.14), on the other hand (8.13) entails that wn(T ) * w(T ) in Hp0 . Hence by (8.6) ∗ ∗ lim inf ϕn(wn(T )) ≥ ϕ (w(T )). (8.23) n→∞ Let us now remind (7.17), which displaying the index n reads wn ∈ ∂ϕn(un) in ∗ Hp0 , a.e. in ]0,T [. This is equivalent to un ∈ ∂ϕn(wn), which clearly entails ∗ Dtϕn(wn) = hun,Dtwni a.e. in 0,T [. (8.24) These three displayed formulae and (8.12) yield Z T Z T ∗ lim sup hun, hn − Dtwni dt = lim sup hun, hni − Dtϕn(wn) dt n→∞ 0 n→∞ 0 Z T ∗ ∗ 0 = lim hun, hni dt − lim inf ϕn(wn(T )) + lim ϕn(w ) n→∞ 0 n→∞ n→∞ Z T Z T (8.25) ∗ ∗ 0 ∗ ≤ hu, hi dt − ϕ (w(T )) + ϕ (w ) = hu, hi − Dtϕ (w) dt 0 0 T (8.17) Z = hu, h − Dtwi dt; 0 thus 0 (un, zn) = (un, hn − Dtwn) →(u, h − Dtw) = (u, z) in Vp ×V . (8.26) π˜+ p PSEUDO-MONOTONE EQUATIONS 2793

As (7.18) is fulﬁlled for any n, (8.11) then yields (8.18). Pointwise-in-Time Solution. The result that we just proved is not completely 0 satisfactory, since a priori the global-in-time operator β : Ws,p(ΩT ) ⊂ Vp → P(Vp) 0 may not be canonically associated to any time-parameterized operator Vp → P(Vp). In other terms, that operator might not be of the form β = αb for any maximal 0 monotone operator α : Vp → P(Vp), see (4.25). A priori β might also exhibit memory. All this is excluded by the next result, which is set in the diﬀerential setup of Sect. 4. We are now able to derive our main result, which provides a pointwise-in-time representation of the limit operator β (see (8.11)). This is based on a number of assumptions on the data, that in particular include the hypotheses of Corollary 4.9 and of the preceding results of this section. Theorem 8.4. Let us prescribe (8.1) and the assumptions of Propositions 8.1 and 8.2. Let S be a family of convex mappings Hp → R such that

∃C1, ..., C4 > 0 : ∀ϕ ∈ S, ∀v ∈ Hp, C kvkp + C ≤ ϕ(v) ≤ C kvkp + C , (8.27) 1 Hp 2 3 Hp 4

∃C5 > 0 : ∀ϕ ∈ S, ∀(v1, w1), (v2, w2) ∈ G(∂ϕ), 2 (8.28) hw1 − w2, v1 − v2i ≥ C5kv1 − v2kH . Let F be a family of L(Ω)⊗B(R×RN )-measurable mappings Ω×R×RN → RN such that ∃a, b > 0 : ∀f ∈ F, for a.e. x ∈ Ω, ∀v ∈ R, ∀ξ ∈ RN , (8.29) f(·, v, ξ)·ξ ≥ a|ξ|p − b, N ∃C1,C2 > 0 : ∀f ∈ F, for a.e. x ∈ Ω, ∀v ∈ R, ∀ξ ∈ R , p−1 (8.30) |f(x, v, ξ)| ≤ C1|ξ| + C2, p N ∀ξ ∈ L (ΩT ) , ∀M > 0, p0 N (8.31) {f(·, v, ξ): f ∈ F, kvkWs,p(ΩT ) ≤ M} is relatively compact in L (ΩT ) . Let us also assume that there exists a mapping γ such that

γ : Vp → R is convex (hence continuous), (8.32)

∀ sequence {vn} in Vp, vn * v, γ(vn) → γ(v) ⇒ vn → v, (8.33) ∀f ∈ F, ∀v ∈ V , for a.e. x, the mapping p (8.34) RN → P(RN ): ξ 7→ f(x, v, ξ) − γ0(ξ) is monotone, and that there exists a mapping ψ such that N 0 ψ :(Hp0 ) → R is convex and Gˆateaux-diﬀerentiable (with derivative ψ ), (8.35) N ∀ sequence {θn} in (Hp0 ) , N N (8.36) θn * θ in (Hp0 ) , ψ(θn) → ψ(θ) ⇒ θn → θ in (Hp0 ) ,

∀ sequence {un}inVp, ∀sequence {fn} in F, 0 setting Fn :=fn(·, un, ∇un), ∀ξn ∈ ψ (Fn) a.e. in Ω, ∀n, Z (8.37) N Fn *F in (Hp0 ) ⇒ lim inf (Fn − F ) · (∇un − ξn) dx ≥ 0. n→∞ Ω Let 0 p0 0 w ∈ Hp0 , hn * h in L (0,T ; Vp), (8.38) and for any n let (un, wn, zn) be a solution of Problem 7.1n. Let the operator β be as in Proposition 8.2, and let the triplet (u, w, z) be as in Theorem 8.3. (The 2794 AUGUSTO VISINTIN convergences (8.13)–(8.15) thus hold, up to extracting subsequences.) 0 Then there exists a strictly generalized pseudo-monotone operator α : Vp → Vp such that 0 β(u) = α(u) in Vp, a.e. in 0,T [. (8.39) Therefore the triplet (u, w, z) solves the system 0 Dtw + z = h in Vp, a.e. in 0,T [, (8.40)

w ∈ ∂ϕ(u) a.e. in ΩT , (8.41) 0 z = α(u) in Vp, a.e. in 0,T [, (8.42) 0 0 w(0) = w in Vp. (8.43) Proof. By Corollary 4.3 the sequence of operators 0 αn(v) := −∇·fn(·, v, ∇v)(∈ Vp) ∀v ∈ Vp, ∀n is a.l.s.c. in the sense of (4.24). By Corollary 6.4 then there exists an operator 0 α : Vp → P(Vp) that is strictly generalized pseudo-monotone (in the sense of (2.19)– (2.22)) and such that I →Γ I (i.e., α G→ α) sequentially in V ×V 0 w.r.t. the topology π . G(αn) G(α) n p p e+ (8.44) On the other hand, by (8.32)–(8.37) we may apply Corollary 4.9. Setting Fn := p0 N fn(·, un, ∇un) a.e. in ΩT for any n, thus there exists F ∈ L (ΩT ) such that

un → u in Vp, p0 N (8.45) Fn → F in L (ΩT ) . Hence, up to extracting subsequences,

un → u in Vp, a.e. in 0,T [, (8.46) p0 N Fn → F in L (Ω) , a.e. in 0,T [. (8.47) Because of the convergences (8.13)–(8.15), by passing to the limit in the approximating equations we get (8.40), with z = −∇·F in D0(Ω), a.e. in ]0,T [. By (8.47) then 0 zn = −∇·Fn → −∇·F = z in Vp, a.e. in 0,T [. (8.48) Therefore (un, zn) = (un, hn − Dtwn) →(u, h − Dtw) = (u, z) π˜+ 0 (8.49) in Vp ×Vp, a.e. in ]0,T [.

By applying (8.44) for a.e. t ∈ 0,T [, we thus get IG(α)(u, z) = 0 a.e. in ]0,T [, that is, (8.42). Finally, (8.16)–(8.19) correspond to (8.40)–(8.43). Remarks. (i) We emphasize that the asymptotic operator α is generalized pseudo- monotone. (ii) As we remarked in Sect. 4, the hypotheses (8.34) and (8.37) are consistent with the occurrence of either horizontal or vertical inflection points in the graph of the (monotone) functions ξ 7→ f(x, v, ξ), for a.e. x and any v. This extends known results, see e.g. [37]. These functions must however be single-valued and injective. (iii) The identification of the limit operator α is a major issue, but is not addressed here. (iv) Techniques and results of the two preceding sections may also be adapted to equations of the form ∇×[f(x, u, ∇×u)] = h in D0(Ω)3, ∀u ∈ V, (8.50) PSEUDO-MONOTONE EQUATIONS 2795 with V defined as in (2.10). (v) Further issues to be addressed include the extension to the asymptotic behavior of other classes of equations, e.g. of the form ∂ϕ(Dtu) + α(u) 3 h. The application of the above techniques to the classical problem of homogenization of composite materials seems also worth of consideration.

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