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

## TL;DR

This paper studies a mathematical model from volcanology involving surface deformation caused by a magma chamber, analyzing both the direct problem of deformation and the inverse problem of cavity detection, with proofs of well-posedness and stability.

## Contribution

It establishes well-posedness of a Neumann boundary value problem for the Lamé system and proves uniqueness and stability in the inverse problem of cavity identification.

## Key findings

- Well-posedness in weighted Sobolev spaces
- Uniqueness of the inverse problem solution
- Stability estimates for cavity reconstruction

## 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 Lam\'e 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.