In this work we study the optimality of increasing stability of the inverse boundary value problem (IBVP) for the Schrödinger equation. The rigorous justification of increasing stability for the IBVP for the 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)

In this paper, we investigate the interior transmission eigenvalue problem for elastic waves propagating outside a sound-soft or a sound-hard obstacle surrounded by an anisotropic layer. This study is motivated by the inverse problem of identifying an object embedded in an inhomogeneous media in the presence of elastic waves. Our analysis of this non-selfadjoint eigenvalue problem relies on the weak formulation of involved boundary value problems and some fundamental tools in functional analysis.

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichletto-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension n ∈ N.

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)

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichletto-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension n ∈ N.