### Abstract

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

Original language | English (US) |
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Pages (from-to) | 2415-2430 |

Number of pages | 16 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 39 |

Issue number | 10 |

DOIs | |

State | Published - Jul 1 2016 |

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### Keywords

- asymptotic expansions
- harmonic functions in the half-space
- single and double layer potentials

### ASJC Scopus subject areas

- Mathematics(all)
- Engineering(all)

### Cite this

*Mathematical Methods in the Applied Sciences*,

*39*(10), 2415-2430. https://doi.org/10.1002/mma.3648