Harnack inequalities in sub-Riemannian settings

In the present thesis, we discuss the main notions of an axiomatic approach for an invariant Harnack inequality. This procedure, originated from techniques for fully nonlinear elliptic operators, has been developed by Di Fazio, Gutiérrez, and Lanconelli in the general settings of doubling Hölder quasi-metric spaces. The main tools of the approach are the so-called double ball property and critical density property: the validity of these properties implies an invariant Harnack inequality. We are mainly interested in the horizontally elliptic operators, i.e. some second order linear degenerate-elliptic operators which are elliptic with respect to the horizontal directions of a Carnot group. An invariant Harnack inequality of Krylov-Safonov type is still an open problem in this context. In the thesis we show how the double ball property is related to the solvability of a kind of exterior Dirichlet problem for these operators. More precisely, it is a consequence of the existence of some suitable interior barrier functions of Bouligand-type. By following these ideas, we prove the double ball property for a generic step two Carnot group. Regarding the critical density, we generalize to the setting of H-type groups some arguments by Gutiérrez and Tournier for the Heisenberg group. We recognize that the critical density holds true in these peculiar contexts by assuming a Cordes-Landis type condition for the coefficient matrix of the operator. By the axiomatic approach, we thus prove an invariant Harnack inequality in H-type groups which is uniform in the class of the coefficient matrices with prescribed bounds for the eigenvalues and satisfying such a Cordes-Landis condition.

Sustainability in buildings and waste sector: metrics, challanges, and opportunities

Nowadays, the scientific community has devoted a consistent effort to the sustainable development of the waste management sector and resource efficiency in building infrastructures. Waste is the fourth largest source sector of emissions and the municipal solid waste management system is considered as the most complex system to manage, due to its diverse composition and fragmentation of producers and responsibilities. Nevertheless, given the deep complexity that characterize the waste management sector, sustainability is still a challenging task. Interestingly, open issues arise when dealing with the sustainability of the waste sector. In this thesis, some recent advances in the waste management sector have been presented. Specifically, through the analysis of four author publications this thesis attempted to fill the gap in the following open issues: (i) the waste collection and generation of waste considering the pillars of sustainability; (ii) the environmental and social analysis in designing building infrastructures; (iv) the role of the waste collection in boosting sustainable systems of waste management; (v) the ergonomics impacts of waste collection. For this purpose, four author publications in international peer – reviewed journals were selected among the wholly author's contributions (i.e., final publication stage).