The authors consider the problem of determining a source term from boundary measurements, in an elliptic problem. The direct and inverse problems are formulated as follows.
Direct problem: Let be a bounded domain in , with sufficiently regular boundary . One considers the Poisson equation
where and are given in and , respectively. Problem (1) admits a unique solution in the functional space , on which the normal trace
is well defined in as a continuous function of . One defines the observation operator
Inverse problem: Given any input data , and a corresponding observation . Can we uniquely determine the source term such that on , where is solution of (1)?
The last two sections of the article are dedicated to the problem of identifying the sources when some a priori information is available: (a) separation of variables is possible and one factor of the product is known (Section 3); or (b) in the case of a domain source of cylindrical geometry, the area of the base is known (Section 4).