Bistable elliptic equations with fractional diffusion

This work concerns the study of bounded solutions to elliptic nonlinear equations with fractional diffusion. More precisely, the aim of this thesis is to investigate some open questions related to a conjecture of De Giorgi about the one-dimensional symmetry of bounded monotone solutions in all space, at least up to dimension 8. This property on 1-D symmetry of monotone solutions for fractional equations was known in dimension n=2. The question remained open for n>2. In this work we establish new sharp energy estimates and one-dimensional symmetry property in dimension 3 for certain solutions of fractional equations. Moreover we study a particular type of solutions, called saddle-shaped solutions, which are the candidates to be global minimizers not one-dimensional in dimensions bigger or equal than 8. This is an open problem and it is expected to be true from the classical theory of minimal surfaces.

Tailoring properties and functionalities of TiO2 and Ag nanoparticles involved in surfaces engineering processes

The functionalization of substrates through the application of nanostructured coatings allows to create new materials, with enhanced properties. In this work, the development of self-cleaning and antibacterial textiles, through the application of TiO2 and Ag based nanostructured coatings was carried out. The production of TiO2 and Ag functionalized materials was achieved both by the classical dip-padding-curing method and by the innovative electrospinning process to obtain nanofibers doped with nano-TiO2 and nano-Ag. In order to optimize the production of functionalized textiles, the study focused on the comprehension of mechanisms involved in the photocatalytic and antibacterial processes and on the real applicability of the products. In particular, a deep investigation on the relationship between nanosol physicochemical characteristics, nanocoating properties and their performances was accomplished. Self-cleaning textiles with optimized properties were obtained by properly purifying and applying commercial TiO2 nanosol while the studies on the photocatalytic mechanism operating in self-cleaning application demonstrated the strong influence of hydrophilic properties and of interaction surface/radicals on final performance. Moreover, a study about the safety in handling of nano-TiO2 was carried out and risk remediation strategies, based on “safety by design” approach, were developed. In particular, the coating of TiO2 nanoparticles by a SiO2 shell was demonstrated to be the best risk remediation strategy in term of biological response and preserving of photoreactivity. The obtained results were confirmed determining the reactive oxygen species production by a multiple approach. Antibacterial textiles for biotechnological applications were also studied and Ag-coated cotton materials, with significant anti-bacterial properties, were produced. Finally, composite nanofibers were obtained merging biopolymer processing and sol-gel techniques. Indeed, electrospun nanofibers embedded with TiO2 and Ag NPs, starting from aqueous keratin based formulation were produced and the photocatalytic and antibacterial properties were assessed. The results confirmed the capability of electrospun keratin nanofibers matrix to preserve nanoparticle properties.