Harmonic Moments and an Inverse Problem for the Heat Equation

The paper is devoted to the solution of the inverse boundary problem for the heat equation. Let Ω be a connected bounded domain in R n (n ≥ 2) with C l (l ≥ 2) boundary Γ. Consider the mixed problem for the heat equation (1.1)        (ρ(x)∂ t − −)u f (t, x) = 0 in (0, +∞) × Ω, u f (t, x) = f (t, x) o n(0 , +∞) × Γ, u f (0, x) = 0 on Ω. The density ρ(x) is a C l+σ , 0 < σ < 1, function on Ω satisfying (1.2) 0 < ρ 1 ≤ ρ(x) ≤ ρ 2 (< +∞). The inverse data used in the paper is a set of normal derivatives ∂u p ∂ν | (0,2)×Γ where u p is the solution of (1.1) with (1.3) f (t, x) = χ(t) p(x). Here χ(t) is a (arbitrary) fixed C ∞ function satisfying 0 ≤ χ(t) ≤ 1 in R, χ(t) = 1 for t ≥ 1 and χ(t) = 0 for t ≤ 1/2. The function p(x) in (1.1) is the boundary value of a harmonic polynomial p(x) (i.e. p = 0). We assume that ∂u p ∂ν | (0,2)×Γ or, more precisely, (1.4) Γ t 0 ∂u p ∂ν (s, x) q(x) dx ds, 0 < t < 2, are given for all sources f of form (1.3) with p ∈ HP m , where HP m = { harmonic polynomial of degree ≤ m }