On Stability for an Inverse Hyperbolic Problem

KiHyun Yun Michigan State University

                       kyun@math.msu.edu

Abstract: In this talk, we focus on the inverse problem of determining the potential q by the Neumann to Dirichlet map Λ_{q} in the wave equation u_{tt} −Δu+qu = 0 in Ω×(0, T) with u(x, 0) = u_{t}(x, 0) = 0. Here, we establish a Hölder type stability with exponent 1 − ϵ as follows: for any small ϵ > 0, there exists β_{0} > 0 such that

∥q_{1} − q_{2}_{L^{∞}(Ω)} ≤ C∥Λ_{q_{1}} − Λ_{q_{2}} _{∗}^{1−ϵ}

when ∥q_{1} − q_{2}_{H^{β}(ℝ^{n})} ≤ M for some β > β_{0}. Here, ∥·_{∗} represents the operator norm.

To mention the previous related work, twenty years ago (1988), using Sylvester and Uhlmann s method that is well known in inverse problems, Ziqi Sun, and Alessandrini and Sylvester obtained 1/3 − ϵ Holder stability

∥q_{1} − q_{2}_{L^{∞}(Ω)} ≤ C∥Λ_{q_{1}} − Λ_{q_{2}} _{∗}^{1/3−ϵ}.

There has been no remarkable improvement in this result, even though Yamamoto made some results under the strong condition.

As it mentioned in the abstract, we have obtained a nearly Lipschitz type stability estimate under the same condition as Sun.