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The div-curl lemma, one of the basic results of the theory of compensated compactness of Murat and Tartar, does not take over to the case in which the two factors two-scale converge in the sense of Nguetseng. A suitable modification of the differential operators however allows for this extension. The argument follows the lines of a well-known paper of F. Murat of 1978, and uses a two-scale extension of the Fourier transform. This result is also extended to time-dependent functions, and is applied...

L'attività di ricerca di chi scrive si è finora indirizzata principalmente verso l'esame dei modelli di transizione di fase, dei modelli di isteresi, e delle relative equazioni non lineari alle derivate parziali. Qui si illustrano brevemente tali problematiche, indicando alcuni degli elementi che le collegano tra di loro. Il lavoro è organizzato come segue. I paragrafi 1, 2, 3 vertono sulle transizioni di fase: si introducono le formulazioni forte e debole del classico modello di Stefan, e si illustrano...

On the basis of Fitzpatrick's variational formulation of maximal monotone relations, and of Nguetseng's two-scale approach to homogenization, scale-transformations have recently been introduced and used for the periodic homogenization of quasilinear P.D.E.s. This note illustrates some basic results of this method.

Flows of the form $D_{t}u+\alpha (u)\ni h$, with $\alpha$ maximal monotone, are here formulated as null-minimization problems via Fitzpatrick's theory. By means of De Giorgi's notion of $\mathrm{\Gamma }$-convergence, we study the compactness and the structural stability of these flows with respect to variations of the source $h$ and of the operator $\alpha$.

To any maximal monotone operator $\alpha :V\to 𝒫(V)$ ($V$ being a real Banach space),in [MR1009594] S. Fitzpatrick associated a lower semicontinuous and convex function $f:V\times V^{\prime }\to ℝ\cup \{+\mathrm{\infty }\}$ such that $$f(v,v^{\prime })\ge ⟨v^{\prime },v⟩\mathit{\hspace{1em}}\forall (v,v^{\prime }),f(v,v^{\prime })=⟨v^{\prime },v⟩\iff v^{\prime }\in \alpha (v).$$ On this basis, in this work two classes of doubly-nonlinear evolutionary equations are formulated as minimization principles: $$D_t\alpha (u)-\mathrm{div}\overrightarrow{\gamma }(\nabla u)\ni h,\alpha (D_{t}u)-\mathrm{div}\overrightarrow{\gamma }(\nabla u)\ni h;$$ here $\alpha$ and $\overrightarrow{\gamma }$ are maximal monotone mappings, and one of them is assumed to be cyclically monotone. For associated initial- and boundary-value problems, existence of a solution is proved, as well as the stability...

Si considerano problemi di controllo ottimale con una dipendenza non lineare tra il controllo e lo stato. Si mostra come in certi casi la continuità di tale dipendenza, quindi la buona posizione nel senso di Tychonov, è connessa alla forma del funzionale costo. In particolare si esamina un problema di Stefan a due fasi con controllo distribuito nel termine di sorgente.

We reformulate and extend G. Nguetseng’s notion of two-scale convergence by means of a variable transformation, and outline some of its properties. We approximate two-scale derivatives, and extend this convergence to spaces of differentiable functions. The two-scale limit of derivatives of bounded sequences in the Sobolev spaces $W^{1,p}\left({\mathbb{R}}^N\right)$, $L_{rot}^2{\left({\mathbb{R}}^3\right)}^3$, $L_{div}^2{\left({\mathbb{R}}^3\right)}^3$ and $W^{2,p}\left({\mathbb{R}}^N\right)$ is then characterized. The two-scale limit behaviour of the potentials of a two-scale convergent sequence of irrotational fields is finally studied.

Si considerano problemi di controllo ottimale con una dipendenza non lineare tra il controllo e lo stato. Si mostra come in certi casi la continuità di tale dipendenza, quindi la buona posizione nel senso di Tychonov, è connessa alla forma del funzionale costo. In particolare si esamina un problema di Stefan a due fasi con controllo distribuito nel termine di sorgente.

In the framework of the theory of nonequilibrium thermodynamics, phase transitions with glass formation in binary alloys are here modelled as a multi-non-linear system of PDEs. A weak formulation is provided for an initial- and boundary-value problem, and existence of a solution is studied. This model is then reformulated as a minimization problem, on the basis of a theory that was pioneered by Fitzpatrick [MR 1009594]. This provides a tool for the analysis of compactness and structural stability...

Hysteresis operators are illustrated, and a weak formulation is studied for an initial- and boundary-value problem associated to the equation $\left({\partial }^2/\partial t^2\right)\left[u+\mathcal{F}\left(u\right)\right]+Au=f$; here $\mathcal{F}$ is a (possibly discontinuous) hysteresis operator, $A$ is a second order elliptic operator, $f$ is a known function. Problems of this sort arise in plasticity, ferromagnetism, ferroelectricity, and so on. In particular an existence result is outlined.

We define and characterize weak and strong in , and other spaces a transformation of variable, extending Nguetseng's definition. We derive several properties, including weak and strong two-scale compactness; in particular we prove two-scale versions of theorems of Ascoli-Arzelà, Chacon, Riesz, and Vitali. We then approximate two-scale derivatives, and define two-scale convergence in spaces of either weakly or strongly differentiable functions. We also derive two-scale versions...