On an elastic model arising from volcanology: An analysis of the direct and inverse problem

Abstract

In this paper we investigate a mathematical model arising from volcanology describing surface deformation effects generated by a magma chamber embedded into Earth's interior and exerting on it a uniform hydrostatic pressure. The modeling assumptions translate mathematically into a Neumann boundary value problem for the classical Lamé system in a half-space with an embedded pressurized cavity. We establish well-posedness of the problem in suitable weighted Sobolev spaces and analyse the inverse problem of determining the pressurized cavity from partial measurements of the displacement field proving uniqueness and stability estimates.where ∇u is the strain tensor, C is the cavity, p > 0 represents the pressure acting on the boundary of the cavity, n is the outer unit normal vector on ∂C and e 3 = (0, 0, 1).The main purpose of this paper is to derive quantitative stability estimates for the inverse problem of identifying the pressurized cavity C from one measurement of the displacement provided on a portion of the boundary of the half-space.In order to address this issue, we first analyse the well-posedness of (1) under the assumption that ∂C is Lipschitz. We highlight that for the well-posedness we can either impose explicitly some decay conditions at infinity for u and ∇u (see, for example, [7]) or, more suitable for our purposes, set the analysis in some weighted Sobolev spaces where the decay conditions are expressed by means of weights. In particular, we will show the well-posedness in this weighted Sobolev space

An Elastic Model for Volcanology

“…Since we want to find quantitative estimates for the solution, we need to utilize quantitative form of these two inequalities. In [37] the authors proved a quantitative version of these two inequalities in H 1 w pR 3 ¡ zCq. For completeness, in the next lines we recall how to prove these results.…”

“…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.…”

“…In Chapter 3, we analyse the elastic model of Section 1.1 and in particular we give solutions to Problem A and B. The results of this chapter are essentially contained in [36] and [37].…”

Analysis of a Model of Elastic Dislocations in Geophysics

“…We next recall the decay estimates satisfied by the Neumann function N in the case of constant coefficients. In particular, it is not difficult to see from Theorem 4.9 in [9] that there exists a positive constant C = C(α 0 , β 0 , M ), such that for all x, y ∈ R 3 − with x = y, N(x, y)…”

A phase-field approach for detecting cavities via a Kohn-Vogelius type functional

“…On the contrary, the inverse problem consists in the identification of the cavity C given C 0 , and g, and making use of the additional boundary measurements represented by the displacement vector f = u N Σ N . It has been proved that uniqueness for cavities detection holds in the class of Lipschitz domains [68,9] while stability estimates (of logarithmic type) have been proved for more regular cavities, precisely assuming a-priori C 1,α regularity, with 0 < α ≤ 1 ( [68]). Similar stability estimates hold also in the case of elastic inclusions ( [69]).…”