ABSTRACT
For fixed ρ, we consider the solution u(f) to u ″ ( x , t ) + A u ( x , t ) = f ( x ) ρ ( x , t ) ( x ∈ Ω , t > 0 ) u ( x , 0 ) = u ′ ( x , 0 ) = 0 ( x ∈ Ω ) , B j u ( x , t ) = 0 ( x ∈ ∂ Ω , t > 0 : 1 ≤ j ≤ m ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187841/ce56260a-d5ea-4a00-8592-3df3cda26043/content/eq1946.tif"/>
where u ′ = ∂ u ∂ t , u ″ = ∂ 2 u ∂ t 2 , Ω ⊂ ℝ r ( r ≥ 1 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187841/ce56260a-d5ea-4a00-8592-3df3cda26043/content/eq1947.tif"/> is a bounded domain with smooth boundary, A is a uniformly symmetric elliptic differential operator of 2m order with t-independent smooth coefficients, Bj (1 ≤ j ≤ m) are boundary differential operators such that the system {A, Bj }1≤j≤m is well-posed. Let {Cj }1≤j≤m be complementary boundary differential operators of {Bj }1≤j≤m. We consider a multidimensional linear inverse problem : for given Γ ⊂ ∂Γ, T > 0 and n ∈ {1,2, …,m}, determine f(x) (x ∈ Γ) from C j u ( f ) ( x , t ) ( x ∈ Γ , 0 < t < T : 1 ≤ j ≤ n ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187841/ce56260a-d5ea-4a00-8592-3df3cda26043/content/eq1948.tif"/> .
By exact controllability based on the Hilbert Uniqueness Method, we systematically establish the uniqueness, the stability and convergence rates of the Tikhonov regularized solutions for this inverse problem in the case of x-independent ρ. In the case of x-dependent ρ, by the Hilbert Uniqueness Method, we reduce our inverse problem to an equation of the second kind which reconstructs f. Our methodology is widely applicable to various equations in mathematical physics.
