INVERSE PROBLEM FOR A PARABOLIC

EQUATION WITH SPACE-PERIODIC BOUNDARY

CONDITIONS BY CARLEMAN ESTIMATE

Jongsung Choi

Department of Mathematical Sciences The University of Tokyo 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

Abstract

In Ω = ∏_{j=1}^{n}]0, L_{j}[ ⊂ ℝ^{n} with L_{j} > 0, 1 ≤ j ≤ n, we consider

(1.1)  \{ \begin{matrix}u_{t}(t, x) = Δu(t, x) − p(x)u(t, x), 0 < t < T, x ∈ Ω, \\ u(0, x) = μ(x), x ∈ Ω, \\ u|_{Γ_{j}^{0}} = u|_{Γ_{j}^{1}} and \frac{∂u}{∂x_{j}} |_{Γ_{j}^{0}} = \frac{∂u}{∂x_{j}} |_{Γ_{j}^{1}} , 1 ≤ j ≤ n. \end{matrix}

We denote by Γ_{j}^{0} and Γ_{j}^{1} the following parts of the boundary ∂Ω:

(1.2)  Γ_{j}^{0} = ∂Ω ∩ {x_{j} = 0}, Γ_{j}^{1} = ∂Ω ∩ {x_{j} = L_{j}},  j = 1, ⋯, n.

We assume that if n = 1, then ω is an arbitrarily fixed subdomain of Ω =]0, L_{1}[, and if n ≥ 2, then ω is an arbitrarily fixed subdomain, containing boundary of Γ_{j}^{i} (i = 0, 1, j = 1, ⋯, n) of Ω, θ ∈ (0, T) is fixed.

Our inverse problem is to determine p = p(x) and μ = μ(x) from the measurement u(p, μ)|_{(0,T)×ω} and u(p, μ)(θ, x), x ∈ Ω. In this paper, we establish a conditional stability estimate for this inverse problem.

The main ingredient in this paper is a Carleman estimate and an argument used by Bukhgeim and Klibanov [BK]. The paper [BK] is an early works establishing the uniqueness in inverse problems, provided that a suitable Carleman estimate is found. Our Carleman estimate in the case of periodic boundary conditions is proved along the idea in Frusikov and Imanuvilov [FI].

References

[BK] A. L. Bukhgeim and M. A. Klibanov, Global uniqueness of a class of multidimensional inverse problems, Sov. Math. Dokl. 24 (1981), 244–247.

[FI] A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series No. 34, Seoul National University, Seoul, Korea, 1996.

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