On the linearized local Calderón problem. (English) Zbl 1198.31003

The authors prove the following localized version of the linear Calderón problem: If \(\Omega\) is a connected bounded open subset of \(\mathbb{R}^n\), \(n\geq2\), with smooth boundary, then the set of products of harmonic functions in \(C^\infty(\overline{\Omega})\) which vanish on a closed proper subset \(\Gamma\subsetneq\partial\Omega\) of the boundary is dense in \(L^1(\Omega)\).
The proof rests on techniques from the microlocal analysis of analytic singularities of distributions, in particular on the Segal-Bargmann transform. In this context the exponential decay of the transform of an \(L^\infty\)-function supported in the half-space is proved using properties of harmonic exponentials and the maximum principle, drawing on ideas from the proof of Kashiwara’s Watermelon Theorem.


31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35R30 Inverse problems for PDEs
78A30 Electro- and magnetostatics
35J10 Schrödinger operator, Schrödinger equation
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35A22 Transform methods (e.g., integral transforms) applied to PDEs
58J32 Boundary value problems on manifolds
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