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

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### Abstract

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.

### Most cited references4

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### The layer potential technique for the inverse conductivity problem

(1996)
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### 4D volcano gravimetry

(2008)
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### Singular Integral Equations

(1989)
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### Author and article information

###### Journal
2015-08-09
1508.02051
10.1002/mma.3648

Roma01.Math
math.AP

Analysis