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Holmgren-John Unique Continuation Theorem for Viscoelastic Systems

In: Time-dependent Problems in Imaging and Parameter Identification

Author

Listed:
  • Maarten V. de Hoop

    (Rice University, Departments of Computational and Applied Mathematics and Earth, Environmental and Planetary Sciences)

  • Ching-Lung Lin

    (National Cheng Kung University, Department of Mathematics)

  • Gen Nakamura

    (Hokkaido University, Department of Mathematics)

Abstract

We consider Holmgren-John’s uniqueness theorem for a partial differential equation with a memory term when the coefficients of the equation are analytic. This is a special case of the general unique continuation property (UCP) for the equation if its coefficients are analytic. As in the case in the absence of a memory term, the Cauchy-Kowalevski theorem is the key to prove this. The UCP is an important tool in the analysis of related inverse problems. A typical partial differential equation with memory term is the equation describing viscoelastic behavior. Here, we prove the UCP for the viscoelastic equation when the relaxation tensor is analytic and allowed to be fully anisotropic.

Suggested Citation

  • Maarten V. de Hoop & Ching-Lung Lin & Gen Nakamura, 2021. "Holmgren-John Unique Continuation Theorem for Viscoelastic Systems," Springer Books, in: Barbara Kaltenbacher & Thomas Schuster & Anne Wald (ed.), Time-dependent Problems in Imaging and Parameter Identification, pages 287-301, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-57784-1_10
    DOI: 10.1007/978-3-030-57784-1_10
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