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Determination of second-order elliptic operators in two dimensions from partial Cauchy data.
Oleg Y Imanuvilov1, Gunther Uhlmann, Masahiro Yamamoto
1Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA.
Summary
This study characterizes coefficients of elliptic operators using partial boundary data. It proves uniqueness for various equations, including conductivity and Schrödinger, by constructing complex geometrical optics solutions.
Area of Science:
- Mathematics
- Partial Differential Equations
- Inverse Problems
Background:
- Inverse boundary value problems are crucial for determining unknown properties of systems from boundary measurements.
- Existing methods often require full boundary data, limiting applicability.
Purpose of the Study:
- To investigate the inverse boundary value problem for general second-order elliptic operators in two dimensions.
- To determine coefficients using only partial Cauchy data on an arbitrary open boundary subset.
- To establish uniqueness results for various important physical models.
Main Methods:
- Development of a complete characterization for coefficients yielding identical partial Cauchy data.
- Construction of novel complex geometrical optics solutions.
- Application of Carleman estimates as a key analytical tool.
Main Results:
- A comprehensive characterization of coefficient sets based on partial Cauchy data is presented.
- Uniqueness results are proven for isotropic and anisotropic conductivity equations.
- Uniqueness is also demonstrated for magnetic Schrödinger and convection-diffusion equations, with specific modulo considerations.
Conclusions:
- Partial Cauchy data is sufficient for unique coefficient determination in several key elliptic equations.
- The developed method offers a powerful approach to solving inverse problems with incomplete data.
- This work advances the understanding of coefficient recovery in partial differential equations.
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