In this paper we investigate a mathematical model arising from volcanology describing surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. The modeling assumptions translate mathematically into a Neumann boundary value problem for the classical Lamé system in a half-space with an embedded pressurized cavity. We establish well-posedness of the problem in suitable weighted Sobolev spaces and analyse the inverse problem of determining the pressurized cavity from partial measurements of the displacement field proving uniqueness and stability estimates.where ∇u is the strain tensor, C is the cavity, p > 0 represents the pressure acting on the boundary of the cavity, n is the outer unit normal vector on ∂C and e 3 = (0, 0, 1).The main purpose of this paper is to derive quantitative stability estimates for the inverse problem of identifying the pressurized cavity C from one measurement of the displacement provided on a portion of the boundary of the half-space.In order to address this issue, we first analyse the well-posedness of (1) under the assumption that ∂C is Lipschitz. We highlight that for the well-posedness we can either impose explicitly some decay conditions at infinity for u and ∇u (see, for example, [7]) or, more suitable for our purposes, set the analysis in some weighted Sobolev spaces where the decay conditions are expressed by means of weights. In particular, we will show the well-posedness in this weighted Sobolev space