Abstract
We introduce a new approach to the anisotropic Calderón problem, based on a map called Poisson embedding that identifies the points of a Riemannian manifold with distributions on its boundary. We give a new uniqueness result for a large class of Calderón type inverse problems for quasilinear equations in the real analytic case. The approach also leads to a new proof of the result of Lassas et al. (Annales de l’ ENS 34(5):771–787, 2001) solving the Calderón problem on real analytic Riemannian manifolds. The proof uses the Poisson embedding to determine the harmonic functions in the manifold up to a harmonic morphism. The method also involves various Runge approximation results for linear elliptic equations.
Highlights
The anisotropic Calderón problem consists in determining a conductivity matrix of a medium, up to a change of coordinates fixing the boundary, from electrical voltage and current measurements on the boundary
We assume that (M, g) and the matrix valued function A and the function B are real analytic. In this case we show that the source-to-solution mapping, even for small data, determines the manifold and the coefficients A and B up to a diffeomorphism and a built in “gauge symmetry” of the problem: Theorem 1.1 Let (M1, g1) and (M2, g2) be compact connected real analytic Riemannian manifolds with mutual boundary and assume that Q j, j = 1, 2, are quasilinear uniformly elliptic operators of the form (4) satisfying (22)–(27)
This principle is implemented by Poisson embedding and by the fact, which we prove, that the metric can be determined from the knowledge of harmonic functions
Summary
The anisotropic Calderón problem consists in determining a conductivity matrix of a medium, up to a change of coordinates fixing the boundary, from electrical voltage and current measurements on the boundary. The Calderón problem reduces to showing that the equality of DN maps implies that the above identification exists In this case we say that the manifolds can be identified by their harmonic functions. The problem of finding the isometry, or conformal mapping if n = 2, from the knowledge of the DN map is known as the geometric Calderón problem This problem has been solved in [28] in the following cases: Theorem Let (M, g) be a compact connected C∞ Riemannian manifold with C∞ boundary. Our main tool for studying the Poisson embedding and constructing the metric from harmonic functions is the Runge approximation property This property allows one to approximate local solutions to an elliptic equation by global solutions. We require specific approximation results for uniformly elliptic operators, and for completeness they will be given in Appendix A together with proofs
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