Analysis of the Kohn Laplacian on the Heisenberg Group and on Cauchy-Riemann Manifolds

The purpose of this study is to analyse the regularity of a differential operator, the Kohn Laplacian, in two settings: the Heisenberg group and the strongly pseudoconvex CR manifolds. The Heisenberg group is defined as a space of dimension 2n+1 with a product. It can be seen in two different ways: as a Lie group and as the boundary of the Siegel UpperHalf Space. On the Heisenberg group there exists the tangential CR complex. From this we define its adjoint and the Kohn-Laplacian. Then we obtain estimates for the Kohn-Laplacian and find its solvability and hypoellipticity. For stating L^p and Holder estimates, we talk about homogeneous distributions. In the second part we start working with a manifold M of real dimension 2n+1. We say that M is a CR manifold if some properties are satisfied. More, we say that a CR manifold M is strongly pseudoconvex if the Levi form defined on M is positive defined. Since we will show that the Heisenberg group is a model for the strongly pseudo-convex CR manifolds, we look for an osculating Heisenberg structure in a neighborhood of a point in M, and we want this structure to change smoothly from a point to another. For that, we define Normal Coordinates and we study their properties. We also examinate different Normal Coordinates in the case of a real hypersurface with an induced CR structure. Finally, we define again the CR complex, its adjoint and the Laplacian operator on M. We study these new operators showing subelliptic estimates. For that, we don't need M to be pseudo-complex but we ask less, that is, the Z(q) and the Y(q) conditions. This provides local regularity theorems for Laplacian and show its hypoellipticity on M.

Steel hopper silos filled with granular material: experimental tests on the tensile resistance of corrugated sheets and on the filling and discharging pressures

This thesis addresses various aspects related to silos, from the strength of some structural parts to internal actions due to grain. Two hopper silo models were mainly studied, so the thesis is divided into two parts. The first part focuses only on the silo cylinder and deals with the collapse of a silo due to failure of the vertical walls. We had the opportunity to access data from a real silo and perform tensile tests on corrugated sheets. The theoretical and experimental resistance of the corrugated sheet forming the silo cylinder was studied. The resistance was then compared with the internal actions due to grain prescribed by various standards. The second part, however, focused on the hopper of a silo in which a load test (loading and unloading of the silo) was performed. Through the test data, an attempt was made to reproduce the pressures normal to the hopper through analytical reasoning. The experimental pressures were then compared with the theoretical pressures predicted by the standards. In addition, with mathematical reasoning, an attempt was made to reproduce the horizontal pressure on the vertical walls of the silo from the experimental normal pressure in the hopper. In fact, the test was related only to the hopper part and not to the silo cylinder.