Analysis of a linear elastic model relative to a small pressurized cavity embedded in the half-space

Motivated by a vulcanological problem, a sound mathematical approach is established for surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. Modeling assumptions translate the problem into classical elasto-static system (homogeneous and isotropic) in an half-space with an embedded pressurized cavity. The boundary conditions are traction-free for the air/crust boundary and uniformly hydrostatic for the chamber boundary. These are complemented with zero-displacement condition at infinity (with decay rate). The well-posedness of the mathematical problem is established and it is provided an appropriate integral formulation for its solution for cavity with general shape. Based on that, assuming that the pressurized cavity is small with respect to the depth, in the thesis is derived rigorously the principal term of the asymptotic expansion for the surface deformation when the ratio "diameter of the cavity and depth" goes to zero. These are new results in the field of the asymptotic analysis since the case of a no null Neumann boundary conditions on the boundary of the cavity and unbounded domains are jointly considered. Moreover a simplified mathematical scalar version of the problem, based on harmonic functions, is analysed. Using again the setting of the integral equations, the well-posedness of the problem is studied. Some new results on asymptotic expansions in unbounded domains contained a small hole with no null Neumann boundary condition are obtained.