Asymptotic expansion for harmonic functions in the half-space with a pressurized cavity

In this paper, we address a simplied version of a problem arising from volcanology. Specifically, as reduced form of the boundary value problem for the Lame 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 the boundary of 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 the boundary of 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 the boundary of C, we recover a simplied representation based on a polarization tensor.