Math 2.0

Partial differential equations Pseudodifferential and microlocal methods Problem

Calderón problem

Builds on Dirichlet-to-Neumann operatorInverse problemDirichlet problem

Problem

Let Ω⊂ℝn\Omega \subset \R^n, n≥2n \ge 2, be a bounded connected open set with smooth boundary. A conductivity is a measurable function γ\gamma on Ω\Omega with c≤γ≤c−1c \le \gamma \le c^{-1} for some constant c>0c > 0. Let Λγ\Lambda_\gamma be the Dirichlet-to-Neumann operator of the equation ∇⋅(γ∇u)=0\nabla \cdot (\gamma \nabla u) = 0 in Ω\Omega. The problem has two parts:

  1. Uniqueness: if Λγ1=Λγ2\Lambda_{\gamma_1} = \Lambda_{\gamma_2}, does it follow that γ1=γ2\gamma_1 = \gamma_2?
  2. Reconstruction: give a method that recovers γ\gamma from Λγ\Lambda_\gamma.

Remarks

This is the model inverse problem behind electrical impedance tomography, where voltages are applied on the surface of a body and the resulting currents are measured. Calderón posed it in his 1980 paper "On an inverse boundary value problem", where he proved uniqueness for the linearized problem near constant conductivities. Sylvester and Uhlmann proved uniqueness for smooth conductivities in dimension n≥3n \ge 3 in 1987, using complex geometrical optics solutions. Astala and Päivärinta proved uniqueness for all bounded measurable conductivities in dimension n=2n = 2 in 2006. In dimension n≥3n \ge 3 uniqueness for general bounded measurable conductivities is open. The problem is severely ill-posed: the stability of γ\gamma with respect to Λγ\Lambda_\gamma is only logarithmic in general.

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Boundary, concept, no evidence levelBoundaryconceptContinuous function, concept, no evidence levelContinuous functionconceptDirichlet-to-Neumann operator, concept, no evidence levelDirichlet-to-Neumann …conceptDomain in the complex plane, concept, no evidence levelDomain in the complex…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptInverse problem, concept, no evidence levelInverse problemconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptHarmonic function, concept, no evidence levelHarmonic functionconceptDirichlet problem, problem, no evidence levelDirichlet problemproblemCalderón problem, problem, no evidence levelCalderón problemproblem
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Boundary concept
  2. Continuous function concept
  3. Dirichlet-to-Neumann operator concept
  4. Domain in the complex plane concept
  5. Euclidean space object
  6. Function concept
  7. Inverse problem concept
  8. Total derivative concept
  9. Function of several variables concept
  10. Vector-valued function concept
  11. Differentiable function of several variables concept
  12. Partial derivative concept
  13. Vector field concept
  14. Gradient concept
  15. Divergence concept
  16. Laplacian concept
  17. Harmonic function concept
  18. Dirichlet problem problem
  19. Calderón problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Electrical_impedance_tomography
About this page
TagR8Y8DDTWBM
URIm2://R8Y8DDTWBM
Kindproblem
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MSC35R30, 35J25, 65N21
StatusThis copy of the site has no build data. Run m2 status R8Y8DDTWBM.
AboutDirichlet-to-Neumann operator, Inverse problem
UsesDirichlet-to-Neumann operator, Inverse problem, Dirichlet problem

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