An inverse source problem for the Lamé system with variable coefficients

Abstract

According to the linear theory of elasticity, we consider a bounded, compressible, and isotropic body whose mechanical behavior is described by the Lamé system with density and Lamé coefficients depending on the space variables. Assuming null surface displacement on the whole boundary, we discuss the inverse problem of determining a body force by observation of surface traction on a suitable sub-boundary along a sufficiently large time interval. Our main results are a Lipschitz stability estimate and a reconstruction formula.

Keywords:

• Linear elasticity
• Exact controllability
• Inverse problems
• 2000 Mathematics Subject Classifications: 35L20
• 35R30
• 74B05

Acknowledgment

The authors thank referees for their valuable comments as well as Professor Luciano Pandolfi for his useful remarks. The first author was partly supported by the Italian MIUR PRIN Research Project Aspetti Teorici e Applicativi di Equazioni a Derivate Parziali and by the Italian MIUR FIRB Research Project Analisi di Equazioni a Derivate Parziali, Lineari e Non Lineari: Aspetti Metodologici, Modellistica, Applicazioni. The third author was partly supported by Grant 15340027 from the Japan Society for the Promotion of Science and Grant 15654015 from the Ministry of Education, Cultures, Sports and Technology.