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On the Backward Uniqueness Property for the Heat Equation in Two-Dimensional Conical Domains

In this article we deal with the backward uniqueness property of the heat equation in conical domains in two spatial dimensions via Carleman inequality techniques. Using a microlocal interpretation of the pseudoconvexity condition, we improve the bounds of Šverák and Li on the minimal angle in which the backward uniqueness property is displayed: We reach angles of slightly less than $95^{\circ}$. Via two-dimensional limiting Carleman weights we obtain the uniqueness of possible controls of the heat equation with lower order perturbations in conical domains with opening angles larger than $90^{\circ}$