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).


35R30 Inverse problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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