Optimality of increasing stability for an inverse boundary value problem

Abstract

In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for Schrödinger equation. The rigorous justification of increasing stability for the IBVP for Schrödinger equation were established by Isakov [Isa11] and by Isakov, Nagayasu, Uhlmann, Wang of the paper [INUW14]. In [Isa11], [INUW14], the authors showed that the stability of this IBVP increases as the frequency increases in the sense that the stability estimate changes from a logarithmic type to a Hölder type. In this work, we prove that the instability changes from an exponential type to a Hölder type when the frequency increases. This result verifies that results in [Isa11], [INUW14] are optimal.(1.2)

The Calderón Problem for the Fractional Wave Equation: Uniqueness and Optimal Stability

“…The estimate (1.11) is shown to be optimal in the recent paper [KUW21]. The estimate (1.11) clearly indicates that the logarithmic part decreases as the frequency κ > 0 increases and the estimate changes from a logarithmic type to a Hölder type.…”

The anisotropic Calderón problem for high fixed frequency

“…where F lead,1 denotes the error term. This is obtained from some perturbation argument in Section 5 and a stationary phase argument by considering the oscillatory behavior in (8) done in Section 6. The error term comes from the application of stationary phase argument.…”

“…This agrees with he phenomena of increased stability for high frequency Schrödinger operators on R n , which has been studied in the literatures. See the recent work [8] and the references therein. Also, we remark that the last inequality in (36) resembles the so-called inverse inequality in numerical methods, see for example [9, Section 6.2].…”