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Gavioli, Chiara, (2021)  - Nuove prospettive in modelli di transizione di fase  - , Tesi di dottorato - (, , Universitą degli studi di Modena e Reggio Emilia ) - pagg. -

Abstract: Phase transitions occur in many relevant processes in physics, natural sciences, and engineering: almost every industrial product involves solidification at some stage. Examples include metal casting, steel annealing, crystal growth, thermal welding, freezing of soil, freezing and melting of the earth surface water, food conservation, and others. All of these processes are characterized by two basic phenomena: heat-diffusion and exchange of latent heat of phase transition. In this thesis, which consists of four distinct parts, we deal with phase transitions from different points of view. The first part, titled "Control and controllability of PDEs with hysteresis with an application in phase transition modeling", is a bridge between controllability of PDEs with hysteresis and phase transitions. Indeed, thanks to the special link between hysteresis operators and phase transitions, the controllability results that we prove can be applied to the so-called relaxed Stefan problem. This is an example of a basic model of phase transition, since it simply accounts for heat-diffusion and exchange of latent heat. More complicated models, which take into account also the mechanical aspects of the process, are considered in the second and in the third part. More precisely in the second part, titled "A viscoelastoplastic porous medium problem with phase transition", we derive and investigate a model for filtration in porous media which takes into account the effects of freezing and melting of water in the pores. The third part, whose title is "Fatigue and phase transition in an oscillating elastoplastic beam", is devoted to the derivation and the study of a model describing fatigue accumulation in an oscillating beam under the hypothesis that the material can partially recover by the effect of melting. Finally, in the fourth part, titled "Regularity for double-phase variational problems", we address the problem of the higher differentiability of solutions to the obstacle problem. In particular we deal with the case of non-standard growth conditions, which includes the so-called double-phase functionals. Such functionals describe the behavior of strongly anisotropic materials whose hardening properties drastically change with the point, hence they exhibit the most dramatic phase transition. The techniques here employed are different from those used in the rest of the thesis, since they rely on the direct methods pertaining to the regularity theory in the field of Calculus of Variations.