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Some remarks on the problem of source identification from boundary measurements. (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 ${ℝ}^{d}$, with sufficiently regular boundary ${\Gamma }$. One considers the Poisson equation

$-{\Delta }u=g\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },\phantom{\rule{1.em}{0ex}}{\gamma }_{0}{u:=u|}_{{\Gamma }}=f,\phantom{\rule{2.em}{0ex}}\left(1\right)$

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 ${H}^{1}\left({\Delta },{\Omega }\right)=\left\{u\in {H}^{1}\left({\Omega }\right);{\Delta }u\in {L}^{2}\left({\Omega }\right)\right\}$, on which the normal trace

${\gamma }_{1}u:=\frac{\partial u}{\partial n}\phantom{\rule{1.em}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}{\Gamma }$

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

$C\left(u\right):={\gamma }_{1}u·$

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