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Weak Structural Stability of Pseudo-Monotone Equations

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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 [ f+1g. 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@'(u) + α(u) 3 h, @' being the subdifferential of a convex lower semicontinuous mapping ', 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@'(u) + α(u) 3 h (Dt := @=@t); (1.1) here @' is the subdifferential of a convex and lower semicontinuous mapping ', 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; ru)(∇·:= div); (1.2) α(u) = r×f(u; r×u)(∇×:= curl): (1.3) 2010 Mathematics Subject Classification. 35K60, 49J40, 58E. Key words and phrases. Pseudo-monotone operators, Γ-convergence, structural stability, vari- ational 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 ', 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 2 V0). (1.4) For instance, β may represent a nonlinear differential operator that acts on func- tions 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 fβng 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 8n and hn ! h in V ; then, extracting subsequences that we label by n0, 0 (i) 9β : V ! P(V ) such that βn0 !τ β; (1.5) (ii) 9u 2 V such that un0 * u in V; (iii) β(u) 3 h: One may interpret this conclusion as the (weak) structural stability of the prob- lem (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 2 A and IA(v) = +1 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 ) 2 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@'(u) + α(u) 3 h in V = L (0;T ; V ) ; 0 (1.7) @'(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 ' is a convex and lower semicontinuous mapping, 1 < p < +1, 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 com- pactness 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 asymptot- ically 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 : V×V0 ! R[f+1g 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 [ f+1g: (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.
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