  Motivated by a vulcanological problem, we establish a sound mathematical
approach 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 cavity. The
boundary conditions are traction-free for the air/crust boundary and uniformly
hydrostatic for the crust/chamber boundary. These are complemented with
zero-displacement condition at infinity (with decay rate).
  After a short presentation of the model and of its geophysical interest, we
establish the well-posedness of the problem and provide an appropriate integral
formulation for its solution for cavity with general shape. Based on that,
assuming that the chamber is centred at some fixed point $\bm{z}$ and has
diameter $\varepsilon>0$, small with respect to the depth $d$, we derive
rigorously the principal term in the asymptotic expansion for the surface
deformation as $\varepsilon=r/d\to 0^+$. Such formula provides a rigorous proof
of the Mogi point source model in the case of spherical cavities generalizing
it to the case of cavities of arbitrary shape.

Analysis of PDEs