On the linearized local Calderón problem

Abstract

Abstract. In this article, we investigate a density problem coming from the linearization of Calderón's problem with partial data. More precisely, we prove that the set of products of harmonic functions on a bounded smooth domain Ω vanishing on any fixed closed proper subset of the boundary are dense in L 1 (Ω) in all dimensions n ≥ 2. This is proved using ideas coming from the proof of Kashiwara's Watermelon theorem [15].

Recent progress in the Calderón problem with partial data

“…Here we sketch the proof of Theorem 2.6. A first reduction is to show that it suffices to prove a local version of the theorem (see [DKSjU09, Section 2]). The global version follows from the local one by using ideas in the spirit of the Runge approximation theorem, developed in an unpublished work of Alessandrini, Isozaki, and Uhlmann.…”

“…Let b, L > 0. Let F be an entire function in C such thate − Φ(s) h |F (s)| ≤ 1, s ∈ C, e − c 2h , when |s − L| ≤ b.Then for all r ≥ 0, there exist c ′ , δ > 0 (independent of F ) such that|F (s)| ≤ e − c ′ 2h + (Im s) 2 2hwhen |Re s| ≤ δ and |Im s| ≤ r.The proof of the lemma rests on a harmonic majorization argument, which exploits the subharmonicity of −(Im s) 2 +(Re s) 2 (see[DKSjU09, Lemma 4.1 and Remark 4.2] for the details of the proof). Next we apply Lemma 7.2 toF (s) = hT f (s, x ′ ) C f L ∞ (Ω) .…”

Stability estimates for the Radon transform with restricted data and applications

“…It served as a substitute to the original but somewhat more involved argument of Kenig, Sjöstrand and Uhlmann in [33] also based on analytic microlocal theory. Similar ideas were used in [20] to investigate a linearization of the Calderón problem with partial data.…”

Local analytic regularity in the linearized Calderón problem

“…(4.29) 4 In the present situation this means that we deform the integration contour so that its boundary remains in the the complex hyperplane y n = 0 and so that the supremum of the modulus of the exponential in the integral is as small as possible, see [17] and Example 4.1 below.…”

“…The same is valid for the case of partial data and the linearization. It was shown in [4] that the linearization of the local DN map at the 0 potential is injective. We consider the linearization of the local DN map at any real analytic potential assuming that the local DN map is measured an an open real-analytic set.…”