Unique continuation property of solutions to general second order elliptic systems
-
Naofumi Honda
and
Satoshi Sasayama
Abstract
This paper concerns the weak unique continuation property of solutions of a general system of differential equation/inequality with a second order strongly elliptic system as its leading part. We put not only some natural assumptions which we call basic assumptions, but also some technical assumptions which we call further assumptions. It is shown as usual by first applying the Holmgren transform to this equation/inequality and then establishing a Carleman estimate for the leading part of the transformed inequality. The Carleman estimate is given via a partition of unity and the Carleman estimate for the operator with constant coefficients obtained by freezing the coefficients of the transformed leading part at a point. A little more details about this are as follows. Factorize this operator with constant coefficients into two first order differential operators. Conjugate each factor by a Carleman weight, and derive an estimate which is uniform with respect to the point at which we froze the coefficients for each conjugated factor by constructing a parametrix for its adjoint operator.
Dedicated to Professor Mikhail Klibanov’s 70th birthday. We admire his distinguished contribution for applying the Carleman estimate to inverse problems. Also, the third author dedicates this paper to the memory of his teacher Prof. Kenjior Okubo.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 15H05740
Award Identifier / Grant number: 19K03554
Funding statement: The second author was partially supported by the Ministry of Science and Technology of Taiwan. The third author acknowledges the support by Grant-in-Aid for Scientific Research of the Japan Society for the Promotion of Science (15H05740 and 19K03554).
Acknowledgements
We thank the anonymous referee for very useful comments which helped improving this paper.
References
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Research biography of a distinguished expert in the field of inverse problems: Professor Michael Victor Klibanov
- Unique continuation property of solutions to general second order elliptic systems
- Calculation of the gradient of Tikhonov’s functional in solving coefficient inverse problems for linear partial differential equations
- Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory
- Alternating direction based method for optimal control problem constrained by Stokes equation
- Direct and inverse scalar scattering problems for the Helmholtz equation in ℝ m
- Mathematical modelling of plasmonic strain sensors
- Finite-dimensional iteratively regularized processes with an a posteriori stopping for solving irregular nonlinear operator equations
- Stability and the inverse gravimetry problem with minimal data
Articles in the same Issue
- Frontmatter
- Research biography of a distinguished expert in the field of inverse problems: Professor Michael Victor Klibanov
- Unique continuation property of solutions to general second order elliptic systems
- Calculation of the gradient of Tikhonov’s functional in solving coefficient inverse problems for linear partial differential equations
- Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory
- Alternating direction based method for optimal control problem constrained by Stokes equation
- Direct and inverse scalar scattering problems for the Helmholtz equation in ℝ m
- Mathematical modelling of plasmonic strain sensors
- Finite-dimensional iteratively regularized processes with an a posteriori stopping for solving irregular nonlinear operator equations
- Stability and the inverse gravimetry problem with minimal data
