TY - JOUR

T1 - Asymptotic expansion for harmonic functions in the half-space with a pressurized cavity

AU - Aspri, Andrea

AU - Beretta, Elena

AU - Mascia, Corrado

N1 - Publisher Copyright:
Copyright © 2016 John Wiley & Sons, Ltd.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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.

AB - 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.

KW - asymptotic expansions

KW - harmonic functions in the half-space

KW - single and double layer potentials

UR - http://www.scopus.com/inward/record.url?scp=84958719333&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958719333&partnerID=8YFLogxK

U2 - 10.1002/mma.3648

DO - 10.1002/mma.3648

M3 - Article

AN - SCOPUS:84958719333

SN - 0170-4214

VL - 39

SP - 2415

EP - 2430

JO - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

IS - 10

ER -