### Abstract

We analyze a mathematical model of elastic dislocations with applications to geophysics, where by an elastic dislocation we mean an open, oriented Lipschitz surface in the interior of an elastic solid, across which there is a discontinuity of the displacement. We model the Earth as an infinite, isotropic, inhomogeneous, elastic medium occupying a half space, and assume only Lipschitz continuity of the Lamé parameters. We study the well posedness of very weak solutions to the forward problem of determining the displacement by imposing traction-free boundary conditions at the surface of the Earth, continuity of the traction and a given jump on the displacement across the fault. We employ suitable weighted Sobolev spaces for the analysis. We utilize the well-posedness of the forward problem and unique-continuation arguments to establish uniqueness in the inverse problem of determining the dislocation surface and the displacement jump from measuring the displacement at the surface of the Earth. Uniqueness holds for tangential or normal jumps and under some geometric conditions on the surface.

Original language | English (US) |
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Journal | Archive for Rational Mechanics and Analysis |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

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### ASJC Scopus subject areas

- Analysis
- Mathematics (miscellaneous)
- Mechanical Engineering

### Cite this

*Archive for Rational Mechanics and Analysis*. https://doi.org/10.1007/s00205-019-01462-w

**Analysis of a Model of Elastic Dislocations in Geophysics.** / Aspri, Andrea; Beretta, Elena; Mazzucato, Anna L.; De Hoop, Maarten V.

Research output: Contribution to journal › Article

*Archive for Rational Mechanics and Analysis*. https://doi.org/10.1007/s00205-019-01462-w

}

TY - JOUR

T1 - Analysis of a Model of Elastic Dislocations in Geophysics

AU - Aspri, Andrea

AU - Beretta, Elena

AU - Mazzucato, Anna L.

AU - De Hoop, Maarten V.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We analyze a mathematical model of elastic dislocations with applications to geophysics, where by an elastic dislocation we mean an open, oriented Lipschitz surface in the interior of an elastic solid, across which there is a discontinuity of the displacement. We model the Earth as an infinite, isotropic, inhomogeneous, elastic medium occupying a half space, and assume only Lipschitz continuity of the Lamé parameters. We study the well posedness of very weak solutions to the forward problem of determining the displacement by imposing traction-free boundary conditions at the surface of the Earth, continuity of the traction and a given jump on the displacement across the fault. We employ suitable weighted Sobolev spaces for the analysis. We utilize the well-posedness of the forward problem and unique-continuation arguments to establish uniqueness in the inverse problem of determining the dislocation surface and the displacement jump from measuring the displacement at the surface of the Earth. Uniqueness holds for tangential or normal jumps and under some geometric conditions on the surface.

AB - We analyze a mathematical model of elastic dislocations with applications to geophysics, where by an elastic dislocation we mean an open, oriented Lipschitz surface in the interior of an elastic solid, across which there is a discontinuity of the displacement. We model the Earth as an infinite, isotropic, inhomogeneous, elastic medium occupying a half space, and assume only Lipschitz continuity of the Lamé parameters. We study the well posedness of very weak solutions to the forward problem of determining the displacement by imposing traction-free boundary conditions at the surface of the Earth, continuity of the traction and a given jump on the displacement across the fault. We employ suitable weighted Sobolev spaces for the analysis. We utilize the well-posedness of the forward problem and unique-continuation arguments to establish uniqueness in the inverse problem of determining the dislocation surface and the displacement jump from measuring the displacement at the surface of the Earth. Uniqueness holds for tangential or normal jumps and under some geometric conditions on the surface.

UR - http://www.scopus.com/inward/record.url?scp=85074838928&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85074838928&partnerID=8YFLogxK

U2 - 10.1007/s00205-019-01462-w

DO - 10.1007/s00205-019-01462-w

M3 - Article

AN - SCOPUS:85074838928

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

ER -