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.

Kinematic description of the rupture from strong motion data: strategies for a robust inversion

We present a non linear technique to invert strong motion records with the aim of obtaining the final slip and rupture velocity distributions on the fault plane. In this thesis, the ground motion simulation is obtained evaluating the representation integral in the frequency. The Green’s tractions are computed using the discrete wave-number integration technique that provides the full wave-field in a 1D layered propagation medium. The representation integral is computed through a finite elements technique, based on a Delaunay’s triangulation on the fault plane. The rupture velocity is defined on a coarser regular grid and rupture times are computed by integration of the eikonal equation. For the inversion, the slip distribution is parameterized by 2D overlapping Gaussian functions, which can easily relate the spectrum of the possible solutions with the minimum resolvable wavelength, related to source-station distribution and data processing. The inverse problem is solved by a two-step procedure aimed at separating the computation of the rupture velocity from the evaluation of the slip distribution, the latter being a linear problem, when the rupture velocity is fixed. The non-linear step is solved by optimization of an L2 misfit function between synthetic and real seismograms, and solution is searched by the use of the Neighbourhood Algorithm. The conjugate gradient method is used to solve the linear step instead. The developed methodology has been applied to the M7.2, Iwate Nairiku Miyagi, Japan, earthquake. The estimated magnitude seismic moment is 2.6326 dyne∙cm that corresponds to a moment magnitude MW 6.9 while the mean the rupture velocity is 2.0 km/s. A large slip patch extends from the hypocenter to the southern shallow part of the fault plane. A second relatively large slip patch is found in the northern shallow part. Finally, we gave a quantitative estimation of errors associates with the parameters.

Caratterizzazione morfologica e neurochimica dei neuroni sensitivi e motori del trigono vescicale e del muscolo uretrale di maiale

Il trigono della vescica urinaria (UBT) è un'area limitata attraverso la quale penetrano nella vescica la maggior parte dei vasi e fibre e in cui le fibre nervose e neuroni intramurali sono più concentrati. Mediante l’utilizzo combinato di un tracciante retrogrado(FB) e dell’immunoistochimica sono stati valutati il fenotipo e l’area del soma dei neuroni dei gangli spinali (DRG), dei neuroni post-gangliari, il fenotipo dei gangli della catena simpatica (STG) e i gangli mesenterici caudali (CMG) innervanti l’UBT. - Caratterizzazione dei neuroni dei DRG con: peptide correlato al gene della calcitonina (CGRP)(30±3%, 29±3%, rispettivamente), sostanza P(SP)(26±8%, 27±12%), ossido nitrico sintasi neuronale (nNOS)(21±4%; 26±7%), neurofilamento 200kDa (NF200)(75±14%, 81±7% ) , transient receptor potential vanilloid1 (TRPV1)(48±13%, 43±6%) e isolectina-B4-positivi (IB4) (56±6%;43±10%). I neuroni sensoriali, distribuiti da L2 a Ca1 (DRG), hanno presentato una localizzazione segmentale, mostrando maggior densità nei DRG L4-L5 e S2-S4. I neuroni sensoriali lombari sono risultati significativamente più grandi di quelle sacrali (1.112±624μm2 vs716±421μm2). Complessivamente, questi dati indicano che le vie lombari e sacrali probabilmente svolgono ruoli diversi nella trasmissione sensitiva del trigono della vescica urinaria. -I neuroni FB+ della STG e dei CMG sono risultati immunoreattivi per la tirosina idrossilasi (TH)(66±10,1%, 53±8,2%, rispettivamente), la dopamina beta-idrossilasi (DβH)(62±6,2%, 52±6,2%), neuropeptideY (NPY)(59±8%; 66±7%), CGRP(24±3%, 22±3%), SP(22±2%; 38±8%), polipeptide intestinale vasoattivo (VIP)(19±2%; 35±4%), nNOS(15±2%; 33±8%), trasportatore vescicolare dell'acetilcolina (VAChT)(15±2%; 35±5%), leu-encefalina (LENK)(14±7%; 26±9%), e somatostatina (SOM)(12±3%;32±7%).Il numero medio di neuroni FB+ (1845,1±259,3) era nella STG in L1-S3, con i pirenofori più piccoli (465,6±82.7μm2). Un gran numero (4287,5±1450,6) di neuroni FB+ di piccole dimensioni (476,1±103,9μm2) sono stati localizzati lungo il margine dei CMG. Il maggior numero (4793,3±1990,8) di neuroni FB + è stato osservato nel plesso pelvico, dove i neuroni marcati erano raggruppati in micro-gangli e con pirenoforo ancora più piccolo (374,9±85,4 μm2).