Asymptotic Expansion for Harmonic Functions in the Half-Space with a Pressurized Cavity

In this paper, we address a simplified version of a problem arising from volcanology. Specifically, as reduced form of the boundary value problem for the Lam\'e system, we consider a Neumann problem for harmonic functions in the half-space with a cavity $C$. Zero normal derivative is assumed at the boundary of the half-space; differently, at $\partial C$, the normal derivative of the function is required to be given by an external datum $g$, corresponding to a pressure term exerted on the medium at $\partial C$. Under the assumption that the (pressurized) cavity is small with respect to the distance from the boundary of the half-space, we establish an asymptotic formula for the solution of the problem. Main ingredients are integral equation formulations of the harmonic solution of the Neumann problem and a spectral analysis of the integral operators involved in the problem. In the special case of a datum $g$ which describes a constant pressure at $\partial C$, we recover a simplified representation based on a polarization tensor.