On the linearized local Calderon problem

In this article, we investigate a density problem coming from the linearization of Calder'on’s problem with partial data. More precisely, we prove that the set of products of harmonic functions on a bounded smooth domain $\mathrm{\Omega }$ vanishing on any fixed closed proper subset of the boundary are dense in $L^1(\mathrm{\Omega })$ in all dimensions $n\ge 2$. This is proved using ideas coming from the proof of Kashiwara’s Watermelon theorem.