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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 Ω be a bounded domain in d , with sufficiently regular boundary Γ. One considers the Poisson equation

-Δu=ginΩ,γ 0 u:=u| Γ =f,(1)

where f and g are given in H 1 2 (Γ) and L 2 (Ω), respectively. Problem (1) admits a unique solution in the functional space H 1 (Δ,Ω)={uH 1 (Ω);ΔuL 2 (Ω)}, on which the normal trace

γ 1 u:=u nonΓ

is well defined in H -1 2 (Γ) as a continuous function of u. One defines the observation operator

C(u):=γ 1 u·

Inverse problem: Given any input data fH 1 2 (Γ), and a corresponding observation φH -1 2 (Γ). Can we uniquely determine the source term g such that C(u)=φ on Γ, 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).

35R30Inverse problems for PDE
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation