Carleman Type Estimates and Their Applications

On increased stability in the continuation of the Helmholtz equation

“…In near future we plan to study this issue theoretically and to link it to the increased stability for the Helmholtz equation and to the (largely open) problem of the exact controllabity in a subdomain. Observe that the exact controllability in the whole domain is relatively well understood [5], [8].…”

Carleman estimate for Maxwell's equations in anisotropic media and the observability inequality

“…The proof of the proposition is done by combining Theorem 3.2 in [4] and Theorem 2.1 in [7] (or [6]), and here we omit the details and give in the succeeding paper [13].…”

Carleman Estimates and Applications to Inverse Problems

“…In other words, (3.2) is a subelliptic estimate for the operator A(·, ∂ − τ ∇ϕ) with a (large) parameter τ . Complete proofs need some technique of (pseudo-)differential quadratic forms and they are given in [6] (for compactly supported u), [16,23]. Now we will give an elementary proof of the Carleman estimate (1.2) for the Helmholtz operator A = −∆ − k 2 .…”

“…Using this reduction and Theorem 4.1 one can obtain uniqueness and stability for the Cauchy problem for the dynamical Lamé system. For more detail and for results on Maxwell's systems and more complicated ones from elasticity theory we refer to [4,8,12,14,16]. Vol.…”