Stability of memory reconstruction from the Dirichlet-to-Neumann operator

Determination of the memory kernel from boundary measurements on a finite time interval

“…In the case T = 1 we problem under consideration was studied in [1], wherein a conditional stability estimate of the logarithmic type was proven. The proof was based on the reduction of the original problem to a family of stationary problems with a parameter by means of the Fourier transform in time.…”

“…The assertion can be derived from the general theory of the initial-boundary value problems for hyperbolic equations as it was done, for example, in [1].…”

Determination of a time‐ and space‐dependent heat flux relaxation function by means of a restricted Dirichlet‐to‐Neumann operator

“…In view of the assumptions Im r j = 0, r j (0; x)¿r 0 ¿0 and the assertions (22), (23) of Lemma 2.3, the functions R j satisfy the inequality R j (p 0 ; x)¿Ä=p 0 ¿0 for x ∈ . Moreover, due to (14) we have R j (p 0 ; ·) ∈ W 2;∞ ( ) ∩ C 1; ( ), j = 1; 2. Let z be an arbitrary point on .…”

“…DNO given on all space-and timedependent Dirichlet data. In particular, Bukhgeim et al [14] uses the Fourier transform to prove the identiÿability.…”

“…References [11][12][13]). For instance, in Reference [14] a problem to identify a memory kernel contained in a lower order term of a hyperbolic equation was studied. In all mentioned papers the time-and space-dependent unknowns are recovered by means of the full DNO, i.e.…”