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.

Maximum principle, mean value operators and quasi boundedness in non-euclidean settings

This work deals with some classes of linear second order partial differential operators with non-negative characteristic form and underlying non- Euclidean structures. These structures are determined by families of locally Lipschitz-continuous vector fields in RN, generating metric spaces of Carnot- Carath´eodory type. The Carnot-Carath´eodory metric related to a family {Xj}j=1,...,m is the control distance obtained by minimizing the time needed to go from two points along piecewise trajectories of vector fields. We are mainly interested in the causes in which a Sobolev-type inequality holds with respect to the X-gradient, and/or the X-control distance is Doubling with respect to the Lebesgue measure in RN. This study is divided into three parts (each corresponding to a chapter), and the subject of each one is a class of operators that includes the class of the subsequent one. In the first chapter, after recalling “X-ellipticity” and related concepts introduced by Kogoj and Lanconelli in [KL00], we show a Maximum Principle for linear second order differential operators for which we only assume a Sobolev-type inequality together with a lower terms summability. Adding some crucial hypotheses on measure and on vector fields (Doubling property and Poincar´e inequality), we will be able to obtain some Liouville-type results. This chapter is based on the paper [GL03] by Guti´errez and Lanconelli. In the second chapter we treat some ultraparabolic equations on Lie groups. In this case RN is the support of a Lie group, and moreover we require that vector fields satisfy left invariance. After recalling some results of Cinti [Cin07] about this class of operators and associated potential theory, we prove a scalar convexity for mean-value operators of L-subharmonic functions, where L is our differential operator. In the third chapter we prove a necessary and sufficient condition of regularity, for boundary points, for Dirichlet problem on an open subset of RN related to sub-Laplacian. On a Carnot group we give the essential background for this type of operator, and introduce the notion of “quasi-boundedness”. Then we show the strict relationship between this notion, the fundamental solution of the given operator, and the regularity of the boundary points.

Sviluppo di un tomografo multi-energy per lo studio pre-clinico di nuove metodiche diagnostiche finalizzate al riconoscimento precoce della patologia tumorale

A new multi-energy CT for small animals is being developed at the Physics Department of the University of Bologna, Italy. The system makes use of a set of quasi-monochromatic X-ray beams, with energy tunable in a range from 26 KeV to 72 KeV. These beams are produced by Bragg diffraction on a Highly Oriented Pyrolytic Graphite crystal. With quasi-monochromatic sources it is possible to perform multi-energy investigation in a more effective way, as compared with conventional X-ray tubes. Multi-energy techniques allow extracting physical information from the materials, such as effective atomic number, mass-thickness, density, that can be used to distinguish and quantitatively characterize the irradiated tissues. The aim of the system is the investigation and the development of new pre-clinic methods for the early detection of the tumors in small animals. An innovative technique, the Triple-Energy Radiography with Contrast Medium (TER), has been successfully implemented on our system. TER consist in combining a set of three quasi-monochromatic images of an object, in order to obtain a corresponding set of three single-tissue images, which are the mass-thickness map of three reference materials. TER can be applied to the quantitative mass-thickness-map reconstruction of a contrast medium, because it is able to remove completely the signal due to other tissues (i.e. the structural background noise). The technique is very sensitive to the contrast medium and is insensitive to the superposition of different materials. The method is a good candidate to the early detection of the tumor angiogenesis in mice. In this work we describe the tomographic system, with a particular focus on the quasi-monochromatic source. Moreover the TER method is presented with some preliminary results about small animal imaging.