## 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 $$\mathbb{R}^d$$, with sufficiently regular boundary $$\Gamma$$. One considers the Poisson equation $-\Delta u= g\quad\text{in }\Omega,\quad \gamma_0 u:= u|_\Gamma= f,\tag{1}$ where $$f$$ and $$g$$ are given in $$H^{{1\over 2}}(\Gamma)$$ and $$L^2(\Omega)$$, respectively. Problem (1) admits a unique solution in the functional space $$H^1(\Delta, \Omega)= \{u\in H^1(\Omega); \Delta u\in L^2(\Omega)\}$$, on which the normal trace $\gamma_1 u:={\partial u\over\partial n}\quad\text{on }\Gamma$ is well defined in $$H^{-{1\over 2}}(\Gamma)$$ as a continuous function of $$u$$. One defines the observation operator $C(u):= \gamma_1u.$ Inverse problem: Given any input data $$f\in H^{{1\over 2}}(\Gamma)$$, and a corresponding observation $$\varphi\in H^{-{1\over 2}}(\Gamma)$$. Can we uniquely determine the source term $$g$$ such that $$C(u)= \varphi$$ 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 PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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