In this seminar I will present a physical model used in volcanology to describe grounds deformations within calderas. Based on the linear elastic theory, this analytical model replaces caldera with a homogeneous, isotropic half-space and the magma chamber by a pressurized cavity of small dimensions compared to its depth. I will exhibit results on the well-posedness of this problem and then the asymptotic formula for the solution within the framework of layer potentials techniques. In the second part of the talk I will show the application of the boundary elements method to a simplified version of the elastic model, in order to discretize the direct problem and generate synthetic data for the study of the inverse problem. This is a joint work with Corrado Mascia and Elena Beretta.