Determination of point sources in vibrating plate by boundary measurements

Uniqueness results in the identification of distributed sources over Germain–Lagrange plates by boundary measurements

“…In [6] Zaïr uses the set (2), but in her work she observes the second derivative of the displacement along the boundary for a longer interval of time.…”

“…For the uniqueness results found in [6], data are collected in a subset of Ω that must contain + ( ) for some ∈ ℝ 2 . Now we define two sets,Γ , + ( 0 ) andΓ , + ( 0 ) , that are going to be our data in the inverse problems analyzed below.…”

“…The method of proof is different from those applied in [5,6]. In fact, the one presented here relies on a kind of comparison between the plate equation and the ordinary wave equation.…”

“…For arbitrary 0 ∈ ℝ 2 , the data were observed along a part of the boundary Γ = { ∈ Ω ∶ ( − 0 ) ⋅ ( ) > 0 } , where ( ) is the outward unit normal to the boundary of Ω, and ( − 0 ) ⋅ ( ) denotes the usual scalar product in ℝ 2 between the vectors ( − 0 ) and ( ). In [6], the observation time must satisfy > Ω, 0 > 0, where Ω, 0 depends on Ω and 0 .…”

“…The identification of point sources was also analyzed in [6]. For arbitrary 0 ∈ ℝ 2 , the data were observed along a part of the boundary Γ = { ∈ Ω ∶ ( − 0 ) ⋅ ( ) > 0 } , where ( ) is the outward unit normal to the boundary of Ω, and ( − 0 ) ⋅ ( ) denotes the usual scalar product in ℝ 2 between the vectors ( − 0 ) and ( ).…”