Lipschitz stability for hyperbolic inequalities in octants with the lateral Cauchy data and refocising in time reversal

Abstract

Hyperbolic equations and inequalities in octants with the lateral Cauchy data at coordinate palnes are consdered.Lipschitz stability estimate is established in the case when both the inhomogeneous right hand side and (unknown) initial conditions at {t = 0} have a finite support. This is the first stability estimate for such a Cauchy problem in an infinite domain. Refocusing of time reversal fields in octants follows. It is shown that the modified Quasi-Reversibility Method can be applied for the numerical solution of such a Cauchy probem including computational time reversal.

A new version of the quasi-reversibility method for the thermoacoustic tomography and a coefficient inverse problem

“…The estimate (1.10) implies a similar estimate for the unknown initial condition [14]. The knowledge of the fact that one of initial conditions was zero was used in [14] for either odd or even extension with respect to t of the function u(x, t) in {t < 0} , depending on which of initial conditions was assumed to be unknown. The proof of [14] is based on the Carleman estimate.…”

“…The knowledge of the fact that one of initial conditions was zero was used in [14] for either odd or even extension with respect to t of the function u(x, t) in {t < 0} , depending on which of initial conditions was assumed to be unknown. The proof of [14] is based on the Carleman estimate. The method of Carleman estimates was first applied in [11] to obtain the Lipschitz stability for the hyperbolic problem with the lateral Cauchy data, also see [10] and Theorem 2.4.1 in [13].…”

“…This leads us to the case T > R. Namely, we want to solve an analogue of the Inverse Problem 1 in a quadrant, assuming that the lateral Cauchy data are given only on parts of two coordinate axis. We are motivated by the publication [14], where the Lipschitz stability was proven for an analogue of Inverse Problem 1 for the case of either a quadrant in R 2 or a an octant in R 3 , assuming that one of initial conditions (1.2) is zero, and the second one has a finite support.…”

Thermoacoustic tomography with an arbitrary elliptic operator

“…In [13] (1.13) was proved for the case of the hyperbolic equation (1.8) with L = ∆+ (variable lower order terms). Next, the result of [13] was extended in [12,15] to a more general case of the hyperbolic inequality…”

“…> 0 and f ∈ L 2 (Q T ) . Although in publications [12,13,15] c ≡ 1, it is clear from them that the key idea is in applying the Carleman estimate, while a specific form of the principal part of the hyperbolic operator is less important. This thought is reflected in the proof of Theorem 3.4.8 of the book [11].…”