The problem of recovering a diffusion coefficient $ a $ in a second-order elliptic partial differential equation from a corresponding solution $ u $ for a given right-hand side $ f $ is considered, with particular focus on the case where $ f $ is allowed to take both positive and negative values. Identifiability of $ a $ from $ u $ is shown under mild smoothness requirements on $ a $, $ f $, and on the spatial domain $ D $, assuming that either the gradient of $ u $ is nonzero almost everywhere, or that $ f $ as a distribution does not vanish on any open subset of $ D $. Further results of this type under essentially minimal regularity conditions are obtained for the case of $ D $ being an interval, including detailed information on the continuity properties of the mapping from $ u $ to $ a $.
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