*(English)*Zbl 0916.35135

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 ${\Omega}$ be a bounded domain in ${\mathbb{R}}^{d}$, with sufficiently regular boundary ${\Gamma}$. One considers the Poisson equation

where $f$ and $g$ are given in ${H}^{\frac{1}{2}}\left({\Gamma}\right)$ and ${L}^{2}\left({\Omega}\right)$, respectively. Problem (1) admits a unique solution in the functional space ${H}^{1}({\Delta},{\Omega})=\{u\in {H}^{1}\left({\Omega}\right);{\Delta}u\in {L}^{2}\left({\Omega}\right)\}$, on which the normal trace

is well defined in ${H}^{-\frac{1}{2}}\left({\Gamma}\right)$ as a continuous function of $u$. One defines the observation operator

Inverse problem: Given any input data $f\in {H}^{\frac{1}{2}}\left({\Gamma}\right)$, and a corresponding observation $\phi \in {H}^{-\frac{1}{2}}\left({\Gamma}\right)$. Can we uniquely determine the source term $g$ such that $C\left(u\right)=\phi $ on ${\Gamma}$, where $u$ 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).

##### MSC:

35R30 | Inverse problems for PDE |

35J05 | Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation |