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

Abstract

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 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.Keywords. Asymptotic expansions; harmonic functions in the half-space; single and double layer potentials. AMS subject classifications. 35C20 (31B10, 35J25)

An Elastic Model for Volcanology

“…where u is the displacement field, C : λI I 2µI is the fourth-order isotropic elasticity tensor with I the identity matrix and I the fourth-order tensor defined by IA : p A; p is a constant representing the pressure. In this book we primarily collect some of the results in [37,35], regarding the well-posedness of problem (1.2) which can be obtained in two different ways: one way is to prescribe the decay conditions at infinity for u, as in (1.2), and then to provide an integral representation formula for u by means of single and double layer potentials of linear elasticity. Then, the well-posedness derives from the study of some integral equations.…”

“…The approach introduced by Ammari and Kang proved to be a very powerful and elegant method to obtain asymptotic expansion of any order for solutions to the transmission problems and, as a particular case, to cavities and perfectly conducting inclusions' problem. This is the reason for which in [35,36] the authors follow the same approach. The mathematical problems contained in these two papers are an intriguing novelty because it is treated the case of a pressurized cavity, i.e., a hole with nonzero tractions on its boundary, buried in an unbounded domain with unbounded boundary.…”

“…The mathematical problems contained in these two papers are an intriguing novelty because it is treated the case of a pressurized cavity, i.e., a hole with nonzero tractions on its boundary, buried in an unbounded domain with unbounded boundary. Some of the results contained in [35,36] are collected in this book.…”

“…As far as we know it doesn't have a physical application but the mathematical result has an interest on its own. Most of the results contained in this chapter come from [35]. In Chapter 3, we analyse the elastic model of Section 1.1 and in particular we give solutions to Problem A and B.…”

Analysis of a Mogi-type model describing surface deformations induced by a magma chamber embedded in an elastic half-space

“…In order to establish these results we use the approach introduced by Ammari and Kang, see for example [4,5,6], based on layer potentials techniques and following the path outlined in [8]. Despite the difficulty to deal with a boundary value problem for an elliptic system in a half-space, see for example [7], this strategy allows us to prove the existence and uniqueness of the displacement field generated by an arbitrary finite cavity C and then to derive the asymptotic expansion (1) as ε goes to zero.…”