Carleman Estimates and Applications to Inverse Problems

Carleman estimates and inverse problems for Dirac operators

“…The equation for h N −1 R N may be solved by Proposition 2.3, which ends the construction of solutions. We still need to consider the solvability of the Eikonal equation (14) and transport equation (15). As discussed in [17], these equations can be solved provided that ϕ is a limiting Carleman weight for the Laplacian, under some geometric assumptions.…”

“…There is an extensive literature on Carleman estimates and their use in unique continuation problems (see for instance [19]). However, Carleman estimates have also become an increasingly useful tool in inverse problems, we refer to [14] for some developments.…”

Recovering a Lamé kernel in a viscoelastic equation by a single boundary measurement

“…We note some related works on inverse problems for hyperbolic equations that make use of Carleman estimates: Bellassoued [1], Bukhgeim [2], Imanuvilov and Yamamoto [8] - [11], Isakov [12], [14], [15], Isakov and Yamamoto [16], Klibanov [17], Klibanov and Timonov [18], Klibanov and Yamamoto [19], Romanov [24], Yamamoto [25] and the references therein. As for available Carleman estimates, see Fursikov and Imanuvilov [5], Hörmander [6], Imanuvilov [7], Isakov [13], Lavrent'ev, V.G.…”

Stability Estimates in Inverse Problems for Hyperbolic Equations

“…If a pseudo-convex function ψ is known, then the following theorem holds (see Theorem 2.1 in [9]). Theorem 3.1.…”

“…On the base of estimate (3.5) one can obtain the Hölder conditional stability estimate for problem (3.1), (3.2) or even a Lipschitz estimate, if T is sufficiently large (see, for instance, [9,12]). Unfortunately, there is no simple way to construct a pseudo-convex function ψ(x, t) for the general operator M inD.…”